Limits of Functions: Definition, Properties and Calculation

What the limit of a function is

The limit of a function describes the behaviour of the values \(f(x)\) as the variable \(x\) approaches a given value, or as \(x\) takes on values that are increasingly large in absolute value.

For instance, writing

\[ \lim_{x \to x_0} f(x)=L \]

means that, as \(x\) approaches \(x_0\), the values of the function \(f(x)\) approach the real number \(L\).

What matters here is that we are not necessarily concerned with the value of the function at the point \(x_0\), but with its behaviour at points near \(x_0\). For this reason the limit may exist even when the function is not defined at \(x_0\), or when \(f(x_0)\) exists but differs from the limit.

In other words, the limit concerns what happens as we approach the point, not what happens exactly at the point. This is what distinguishes the notion of limit from the mere evaluation of \(f(x_0)\).

The same idea applies when the variable does not tend to a real number, but to infinity. Writing

\[ \lim_{x \to +\infty} f(x)=L \]

means that the values \(f(x)\) approach \(L\) as \(x\) takes on positive values that are increasingly large.

The concept of limit therefore allows us to study the local behaviour of a function near a point, together with its global behaviour for very large values of the variable. Fundamental notions of analysis, such as continuity, asymptotes and differential calculus, all rest upon it.

Accumulation points and the meaning of \(x \to x_0\)

Before giving a rigorous definition of limit, we must clarify what it means to say that \(x\) tends to a point \(x_0\).

Let \(f:A\to\mathbb{R}\) be a real-valued function of a real variable, defined on a set \(A\subseteq\mathbb{R}\). When we write

\[ x \to x_0 \]

we are not saying that \(x\) equals \(x_0\), but that \(x\) takes on values in the domain \(A\) arbitrarily close to \(x_0\), possibly distinct from \(x_0\).

For this idea to make sense, the point \(x_0\) must be an accumulation point of the domain of the function. This means that every neighbourhood of \(x_0\) contains at least one point of \(A\) other than \(x_0\).

Equivalently, \(x_0\) is an accumulation point of \(A\) if, for every \(\delta >0\), there exists at least one point \(x\in A\), with \(x\neq x_0\), such that

\[ |x-x_0|<\delta. \]

The condition \(x\neq x_0\) is essential: in studying the limit we are interested in the behaviour of the function at points close to \(x_0\), not necessarily in the value of the function at \(x_0\) itself.

For this reason, \(x_0\) need not belong to the domain \(A\). If, however, there exist points of \(A\) arbitrarily close to \(x_0\), then it is meaningful to study the limit of \(f(x)\) as \(x\to x_0\).

Conversely, if \(x_0\) is an isolated point of the domain, there are no points of the domain arbitrarily close to \(x_0\) and distinct from it. In that case, the limit as \(x\to x_0\) does not describe any genuine approach of the function.

In short, the expression \(x\to x_0\) must always be interpreted with respect to the domain of the function: the variable \(x\) approaches \(x_0\) while taking values for which \(f(x)\) is defined.

Finite limit as \(x\) tends to a finite point

Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\), and let \(x_0\in\mathbb{R}\) be an accumulation point of \(A\).

To say that \(f(x)\) tends to the real number \(L\) as \(x\) tends to \(x_0\) means that the values \(f(x)\) can be made arbitrarily close to \(L\), provided \(x\) is sufficiently close to \(x_0\), with \(x\neq x_0\).

In symbols, we write:

\[ \lim_{x\to x_0} f(x)=L. \]

The rigorous definition is as follows:

\[ \lim_{x\to x_0} f(x)=L \]

if and only if, for every \(\varepsilon >0\), there exists a \(\delta >0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies |f(x)-L|<\varepsilon. \]

The number \(\varepsilon\) measures how close we require \(f(x)\) to be to \(L\). The definition demands that this closeness be attainable for every choice of \(\varepsilon >0\), however small.

The number \(\delta\), on the other hand, measures how close \(x\) must be to \(x_0\) in order that \(f(x)\) be close to \(L\). In general, \(\delta\) may depend on \(\varepsilon\): the smaller the required tolerance on the values of \(f(x)\), the more it may be necessary to shrink the neighbourhood of \(x_0\).

The condition

\[ 0<|x-x_0|<\delta \]

states that \(x\) lies in a neighbourhood of \(x_0\), yet is distinct from \(x_0\). For this reason the value \(f(x_0)\), even if it exists, plays no part in the definition of the limit.

Consequently, the limit may exist even if the function is not defined at \(x_0\). It may also happen that \(f(x_0)\) is defined but differs from the limit. In either case, the limit describes the behaviour of the function at points close to \(x_0\), not necessarily the value it takes at \(x_0\) itself.

Consider, for example, the function

\[ f(x)=\frac{x^2-1}{x-1} \]

which is not defined at \(x=1\). Nevertheless, for \(x\neq 1\), we may simplify:

\[ \frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1. \]

Hence, as \(x\) approaches \(1\), the values of the function approach \(2\). We therefore write:

\[ \lim_{x\to 1}\frac{x^2-1}{x-1}=2. \]

This example shows why, in the study of limits, it is essential to distinguish the behaviour of the function near a point from the value the function takes at that very point.

Infinite limit as \(x\) tends to a finite point

Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\), and let \(x_0\in\mathbb{R}\) be an accumulation point of \(A\).

It may happen that, as \(x\) approaches \(x_0\), the values \(f(x)\) do not approach a real number but instead grow arbitrarily large in absolute value. In this case we speak of an infinite limit.

Writing

\[ \lim_{x\to x_0} f(x)=+\infty \]

means that the values of the function exceed any prescribed positive number, provided \(x\) is sufficiently close to \(x_0\), with \(x\neq x_0\).

The rigorous definition is as follows:

\[ \lim_{x\to x_0} f(x)=+\infty \]

if and only if, for every \(M>0\), there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies f(x)>M. \]

Similarly, writing

\[ \lim_{x\to x_0} f(x)=-\infty \]

means that the values of the function fall below any prescribed negative number, however large in absolute value, provided \(x\) is sufficiently close to \(x_0\), with \(x\neq x_0\).

Formally:

\[ \lim_{x\to x_0} f(x)=-\infty \]

if and only if, for every \(M>0\), there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies f(x)<-M. \]

It is important to observe that \(+\infty\) and \(-\infty\) are not real numbers. To say that a function tends to \(+\infty\) or to \(-\infty\) does not therefore mean that it approaches some numerical value, but rather that its values increase or decrease without bound.

Consider, for example, the function

\[ f(x)=\frac{1}{(x-1)^2}. \]

It is not defined at \(x=1\). Nevertheless, as \(x\) approaches \(1\), the denominator \((x-1)^2\) becomes positive and increasingly close to \(0\). Consequently, the ratio becomes positive and arbitrarily large.

Hence:

\[ \lim_{x\to 1}\frac{1}{(x-1)^2}=+\infty. \]

Likewise, for the function

\[ g(x)=-\frac{1}{(x-1)^2} \]

we have:

\[ \lim_{x\to 1}\left(-\frac{1}{(x-1)^2}\right)=-\infty. \]

Infinite limits are closely connected with vertical asymptotes. If a function tends to \(+\infty\) or to \(-\infty\) as \(x\) tends to \(x_0\), then the vertical line \(x=x_0\) is a vertical asymptote for the graph of the function.

Finite limit as \(x\) tends to infinity

So far we have considered the behaviour of a function as \(x\) approaches a finite point \(x_0\). We may equally well study what happens as \(x\) takes on increasingly large values, or increasingly small ones.

Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\). To study the limit as \(x\to+\infty\), the domain \(A\) must contain arbitrarily large values. In other words, for every real number \(R\) there must exist at least one \(x\in A\) such that \(x>R\).

Writing

\[ \lim_{x\to+\infty} f(x)=L \]

means that the values \(f(x)\) approach the real number \(L\) as \(x\) becomes increasingly large.

The rigorous definition is as follows:

\[ \lim_{x\to+\infty} f(x)=L \]

if and only if, for every \(\varepsilon>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x>R \implies |f(x)-L|<\varepsilon. \]

The meaning here is analogous to that of the \(\varepsilon\)-\(\delta\) definition: the number \(\varepsilon\) fixes how close we require \(f(x)\) to be to \(L\), while the number \(R\) indicates from which point onwards this closeness is guaranteed.

Similarly, to study the limit as \(x\to-\infty\), the domain \(A\) must contain arbitrarily small values. Writing

\[ \lim_{x\to-\infty} f(x)=L \]

means that the values \(f(x)\) approach the real number \(L\) as \(x\) becomes increasingly small.

Formally:

\[ \lim_{x\to-\infty} f(x)=L \]

if and only if, for every \(\varepsilon>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x<R \implies |f(x)-L|<\varepsilon. \]

In this definition, \(R\) is chosen so that, for values of \(x\) less than \(R\), the function takes values close to \(L\).

Consider, for example, the function

\[ f(x)=\frac{1}{x}. \]

As \(x\) takes on increasingly large positive values, the ratio \(\displaystyle \frac{1}{x}\) becomes increasingly close to \(0\). Hence:

\[ \lim_{x\to+\infty}\frac{1}{x}=0. \]

The same occurs as \(x\) takes on increasingly small negative values: the absolute value of \(\displaystyle \frac{1}{x}\) again becomes increasingly small. Thus:

\[ \lim_{x\to-\infty}\frac{1}{x}=0. \]

A finite limit as \(x\to+\infty\) or as \(x\to-\infty\) therefore expresses the fact that, as we move indefinitely along the real axis, the function approaches a fixed real value. This behaviour underlies the notion of a horizontal asymptote.

Infinite limit as \(x\) tends to infinity

We may finally consider the case in which the variable \(x\) tends to infinity while, at the same time, the values of the function also become arbitrarily large or arbitrarily small.

Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\), and suppose that the domain \(A\) contains arbitrarily large values.

Writing

\[ \lim_{x\to+\infty} f(x)=+\infty \]

means that the values of the function exceed any prescribed positive number, provided \(x\) is sufficiently large.

The rigorous definition is as follows:

\[ \lim_{x\to+\infty} f(x)=+\infty \]

if and only if, for every \(M>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x>R \implies f(x)>M. \]

Similarly, writing

\[ \lim_{x\to+\infty} f(x)=-\infty \]

means that the values of the function fall below any prescribed negative number, however large in absolute value, provided \(x\) is sufficiently large.

Formally:

\[ \lim_{x\to+\infty} f(x)=-\infty \]

if and only if, for every \(M>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x>R \implies f(x)<-M. \]

The definitions for \(x\to-\infty\) are analogous. If the domain \(A\) contains arbitrarily small values, then

\[ \lim_{x\to-\infty} f(x)=+\infty \]

if and only if, for every \(M>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x<R \implies f(x)>M. \]

Likewise,

\[ \lim_{x\to-\infty} f(x)=-\infty \]

if and only if, for every \(M>0\), there exists a real number \(R\) such that, for every \(x\in A\),

\[ x<R \implies f(x)<-M. \]

For example, for the function \(f(x)=x^2\) we have:

\[ \lim_{x\to+\infty}x^2=+\infty \]

and also

\[ \lim_{x\to-\infty}x^2=+\infty. \]

Indeed, as \(x\) becomes increasingly large in absolute value, the square \(x^2\) becomes arbitrarily large.

For the function \(g(x)=x^3\), on the other hand, we have:

\[ \lim_{x\to+\infty}x^3=+\infty \]

whereas

\[ \lim_{x\to-\infty}x^3=-\infty. \]

Here the sign of the values of the function depends on the sign of \(x\), since the power has an odd exponent.

Infinite limits as \(x\to+\infty\) or as \(x\to-\infty\) thus describe functions that do not approach a finite real value, but instead increase or decrease without bound along one direction of the real axis.

Right-hand and left-hand limits

In studying the limit of a function as \(x\to x_0\), the variable \(x\) may approach \(x_0\) from two different directions: from the right, or from the left.

To say that \(x\) tends to \(x_0\) from the right means that \(x\) approaches \(x_0\) while taking values greater than \(x_0\). In symbols we write:

\[ x\to x_0^+. \]

To say instead that \(x\) tends to \(x_0\) from the left means that \(x\) approaches \(x_0\) while taking values less than \(x_0\). In symbols we write:

\[ x\to x_0^-. \]

Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\). To study the right-hand limit at \(x_0\), there must exist points of the domain \(A\) arbitrarily close to \(x_0\) and greater than \(x_0\).

Writing

\[ \lim_{x\to x_0^+} f(x)=L \]

means that the values \(f(x)\) approach \(L\) as \(x\) approaches \(x_0\) while taking values greater than \(x_0\).

Formally:

\[ \lim_{x\to x_0^+} f(x)=L \]

if and only if, for every \(\varepsilon>0\), there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<x-x_0<\delta \implies |f(x)-L|<\varepsilon. \]

Similarly, to study the left-hand limit at \(x_0\), there must exist points of the domain \(A\) arbitrarily close to \(x_0\) and less than \(x_0\).

Writing

\[ \lim_{x\to x_0^-} f(x)=L \]

means that the values \(f(x)\) approach \(L\) as \(x\) approaches \(x_0\) while taking values less than \(x_0\).

Formally:

\[ \lim_{x\to x_0^-} f(x)=L \]

if and only if, for every \(\varepsilon>0\), there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<x_0-x<\delta \implies |f(x)-L|<\varepsilon. \]

The conditions \(0<x-x_0<\delta\) and \(0<x_0-x<\delta\) indicate, respectively, that \(x\) lies in a right- or left-hand neighbourhood of \(x_0\), excluding the point \(x_0\) itself.

When \(x_0\) is an accumulation point of the domain both from the left and from the right, the limit as \(x\to x_0\) exists if and only if both the right-hand and left-hand limits exist and are equal. In that case, their common value is the limit of the function at \(x_0\).

In symbols:

\[ \lim_{x\to x_0} f(x)=L \]

if and only if

\[ \lim_{x\to x_0^-} f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0^+} f(x)=L. \]

If, on the other hand, the right-hand and left-hand limits exist but differ, then the limit of the function as \(x\to x_0\) does not exist.

Consider, for example, the function

\[ f(x)=\frac{|x|}{x}. \]

For \(x>0\) we have \(|x|=x\), so that \(f(x)=1\). For \(x<0\), on the other hand, \(|x|=-x\), so that \(f(x)=-1\). Hence:

\[ \lim_{x\to 0^+}\frac{|x|}{x}=1 \]

whereas

\[ \lim_{x\to 0^-}\frac{|x|}{x}=-1. \]

Since the right-hand and left-hand limits differ, the limit

\[ \lim_{x\to 0}\frac{|x|}{x} \]

does not exist.

Uniqueness of the limit

A function cannot have two distinct limits at the same point, or for the same mode of convergence. This fact is expressed by the following theorem.

Theorem (uniqueness of the limit). Let \(f:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\), and let \(x_0\) be an accumulation point of \(A\). If the limits

\[ \lim_{x\to x_0}f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0}f(x)=M \]

both exist, then necessarily

\[ L=M. \]

Proof. Suppose, for a contradiction, that \(L\neq M\). Without loss of generality, we may assume \(L<M\).

Choose

\[ \varepsilon=\frac{M-L}{2}. \]

Since

\[ \lim_{x\to x_0}f(x)=L, \]

there exists \(\delta_1>0\) such that

\[ 0<|x-x_0|<\delta_1 \implies |f(x)-L|<\varepsilon. \]

Likewise, since

\[ \lim_{x\to x_0}f(x)=M, \]

there exists \(\delta_2>0\) such that

\[ 0<|x-x_0|<\delta_2 \implies |f(x)-M|<\varepsilon. \]

Set

\[ \delta=\min\{\delta_1,\delta_2\}. \]

Since \(x_0\) is an accumulation point of \(A\), there exists at least one \(x\in A\) such that

\[ 0<|x-x_0|<\delta. \]

For such an \(x\), both inequalities hold simultaneously:

\[ |f(x)-L|<\varepsilon \qquad\text{and}\qquad |f(x)-M|<\varepsilon. \]

From the first it follows that

\[ L-\varepsilon<f(x)<L+\varepsilon, \]

while from the second we obtain

\[ M-\varepsilon<f(x)<M+\varepsilon. \]

Substituting \(\varepsilon=\displaystyle\frac{M-L}{2}\), we find

\[ L+\varepsilon = L+\frac{M-L}{2} = \frac{L+M}{2}, \]

and likewise

\[ M-\varepsilon = M-\frac{M-L}{2} = \frac{L+M}{2}. \]

Consequently,

\[ f(x)<\frac{L+M}{2} \qquad\text{and}\qquad f(x)>\frac{L+M}{2}, \]

which is impossible.

The assumption \(L\neq M\) therefore leads to a contradiction. It follows that necessarily

\[ L=M. \]

Remark

The theorem guarantees that, whenever a limit exists, it is unique. If instead the right-hand and left-hand limits differ, the limit does not exist, as we saw in the preceding section.

The sign-permanence theorem

The sign-permanence theorem states that if a function tends to a positive limit, then it is positive throughout a sufficiently small neighbourhood of the point considered. Likewise, if it tends to a negative limit, then it is negative throughout a sufficiently small neighbourhood.

This result is important because it allows us to transfer, at least locally, the sign of the limit to the values of the function.

Theorem. Let \(f:A\to\mathbb{R}\) be a function and let \(x_0\) be an accumulation point of \(A\). Suppose that

\[ \lim_{x\to x_0} f(x)=L. \]

If \(L>0\), then there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies f(x)>0. \]

If instead \(L<0\), then there exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies f(x)<0. \]

Proof in the case \(L>0\)

Suppose that \(L>0\). Since

\[ \lim_{x\to x_0} f(x)=L, \]

we may apply the definition of limit, choosing

\[ \varepsilon=\frac{L}{2}. \]

Since \(L>0\), we have \(\varepsilon>0\). There thus exists a \(\delta>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta \implies |f(x)-L|<\frac{L}{2}. \]

From the inequality

\[ |f(x)-L|<\frac{L}{2} \]

it follows in particular that

\[ -\frac{L}{2}<f(x)-L<\frac{L}{2}. \]

Adding \(L\) throughout gives

\[ \frac{L}{2}<f(x)<\frac{3L}{2}. \]

In particular, since \(L>0\), we obtain

\[ f(x)>0. \]

Hence \(f(x)\) is positive at every point of the domain sufficiently close to \(x_0\), with the possible exception of \(x_0\) itself.

The case \(L<0\)

The case \(L<0\) is proved analogously. Choose

\[ \varepsilon=-\frac{L}{2}, \]

which is positive because \(L<0\). From the definition of limit we obtain, for \(x\) sufficiently close to \(x_0\),

\[ |f(x)-L|<-\frac{L}{2}. \]

This inequality implies that \(f(x)\) remains close to the negative number \(L\). More precisely, we obtain

\[ \frac{3L}{2}<f(x)<\frac{L}{2}. \]

Since \(L<0\), the quantity \(\displaystyle\frac{L}{2}\) is also negative. Consequently

\[ f(x)<0. \]

Remarks

The theorem does not assert that the function has the same sign as the limit throughout its entire domain, but only within a sufficiently small neighbourhood of the point towards which the variable tends.

Moreover, if the limit equals zero, no permanence of sign can be deduced. A function may tend to \(0\) while taking positive values, negative values, or values of alternating sign.

The same ideas hold for limits as \(x\to+\infty\) and as \(x\to-\infty\): if the limit is positive, the function is positive from some point onwards; if the limit is negative, the function is negative from some point onwards.

The squeeze theorem

The squeeze theorem (also known as the sandwich theorem) allows us to determine the limit of a function by comparing it with two functions whose limits are already known.

The idea is simple: if a function \(g(x)\) is trapped between two functions \(f(x)\) and \(h(x)\), and if \(f(x)\) and \(h(x)\) tend to the same limit \(L\), then \(g(x)\) must also tend to \(L\).

Theorem. Let \(f,g,h:A\to\mathbb{R}\) be three functions and let \(x_0\) be an accumulation point of \(A\). Suppose there exists a deleted neighbourhood of \(x_0\) throughout which

\[ f(x)\leq g(x)\leq h(x). \]

Suppose further that

\[ \lim_{x\to x_0}f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0}h(x)=L. \]

Then the limit of \(g(x)\) as \(x\to x_0\) also exists, and

\[ \lim_{x\to x_0}g(x)=L. \]

Proof. Fix a number \(\varepsilon>0\). Since

\[ \lim_{x\to x_0}f(x)=L, \]

there exists a number \(\delta_1>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta_1 \implies |f(x)-L|<\varepsilon. \]

From this inequality it follows in particular that

\[ L-\varepsilon<f(x)<L+\varepsilon. \]

Since

\[ \lim_{x\to x_0}h(x)=L, \]

there exists a number \(\delta_2>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta_2 \implies |h(x)-L|<\varepsilon. \]

From this inequality it follows in particular that

\[ L-\varepsilon<h(x)<L+\varepsilon. \]

By hypothesis, there also exists a number \(\delta_0>0\) such that, for every \(x\in A\),

\[ 0<|x-x_0|<\delta_0 \implies f(x)\leq g(x)\leq h(x). \]

Set

\[ \delta=\min\{\delta_0,\delta_1,\delta_2\}. \]

If \(x\in A\) and \(0<|x-x_0|<\delta\), then the following hold simultaneously:

\[ L-\varepsilon<f(x), \qquad f(x)\leq g(x)\leq h(x), \qquad h(x)<L+\varepsilon. \]

Consequently,

\[ L-\varepsilon<g(x)<L+\varepsilon. \]

This is equivalent to

\[ |g(x)-L|<\varepsilon. \]

We have thus shown that, for every \(\varepsilon>0\), there exists a \(\delta>0\) such that

\[ 0<|x-x_0|<\delta \implies |g(x)-L|<\varepsilon. \]

By the definition of limit, it follows that

\[ \lim_{x\to x_0}g(x)=L. \]

Example. Consider the limit

\[ \lim_{x\to 0}x^2\sin\frac{1}{x}. \]

The function \(\displaystyle \sin\frac{1}{x}\) has no limit as \(x\to 0\), since it oscillates indefinitely. Nevertheless, we know that, for every \(x\neq 0\),

\[ -1\leq \sin\frac{1}{x}\leq 1. \]

Multiplying throughout by \(x^2\), which is non-negative, we obtain

\[ -x^2\leq x^2\sin\frac{1}{x}\leq x^2. \]

Since

\[ \lim_{x\to 0}(-x^2)=0 \qquad\text{and}\qquad \lim_{x\to 0}x^2=0, \]

the squeeze theorem gives

\[ \lim_{x\to 0}x^2\sin\frac{1}{x}=0. \]

Comparison with infinite limits

The squeeze theorem also has useful versions for infinite limits.

If, throughout a deleted neighbourhood of \(x_0\),

\[ f(x)\leq g(x) \]

and

\[ \lim_{x\to x_0}f(x)=+\infty, \]

then

\[ \lim_{x\to x_0}g(x)=+\infty. \]

Indeed, if \(f(x)\) exceeds any number \(M>0\), then \(g(x)\), being greater than or equal to \(f(x)\), likewise exceeds \(M\).

Similarly, if, throughout a deleted neighbourhood of \(x_0\),

\[ g(x)\leq h(x) \]

and

\[ \lim_{x\to x_0}h(x)=-\infty, \]

then

\[ \lim_{x\to x_0}g(x)=-\infty. \]

These versions express the same principle: a function forced, locally, to lie above a quantity tending to \(+\infty\) itself tends to \(+\infty\); a function forced to lie below a quantity tending to \(-\infty\) itself tends to \(-\infty\).

Remarks

The comparison need only hold in a neighbourhood of the point considered, possibly excluding the point itself. It is not necessary that the inequalities hold throughout the entire domain of the function.

The squeeze theorem is particularly useful when the function whose limit is sought contains a factor that oscillates but remains bounded, as with the sine and cosine functions.

Operations on limits

The operations on limits allow us to compute the limit of functions obtained through sums, products, quotients and powers, starting from limits already known.

Let \(f,g:A\to\mathbb{R}\), with \(A\subseteq\mathbb{R}\), and let \(x_0\) be an accumulation point of \(A\). Suppose that two finite limits exist:

\[ \lim_{x\to x_0}f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0}g(x)=M. \]

Then the following properties hold.

Limit of a sum

The limit of a sum equals the sum of the limits:

\[ \lim_{x\to x_0}\bigl(f(x)+g(x)\bigr)=L+M. \]

Similarly,

\[ \lim_{x\to x_0}\bigl(f(x)-g(x)\bigr)=L-M. \]

Limit of a product

The limit of a product equals the product of the limits:

\[ \lim_{x\to x_0}\bigl(f(x)g(x)\bigr)=LM. \]

In particular, if \(c\in\mathbb{R}\), then

\[ \lim_{x\to x_0}cf(x)=cL. \]

Limit of a quotient

If \(M\neq 0\), then the limit of a quotient equals the quotient of the limits:

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac{L}{M}. \]

The condition \(M\neq 0\) is essential. Indeed, if the limit of the denominator is non-zero, then, by the sign-permanence theorem, the function \(g(x)\) is non-zero throughout a deleted neighbourhood of \(x_0\). Within that neighbourhood the quotient is therefore well defined.

Limit of a power

If \(n\in\mathbb{N}\), then

\[ \lim_{x\to x_0}\bigl(f(x)\bigr)^n=L^n. \]

This property follows from the limit of a product, applied repeatedly.

Limit of a root

For roots one must pay attention to the domain. If \(\sqrt[n]{f(x)}\) is defined throughout a deleted neighbourhood of \(x_0\), then, whenever the real \(n\)th root is well defined,

\[ \lim_{x\to x_0}\sqrt[n]{f(x)}=\sqrt[n]{L}. \]

In particular, for roots of even index it is necessary that the values considered be non-negative and that the limit \(L\) itself be non-negative.

Examples

Let us compute the limit

\[ \lim_{x\to 2}(3x^2-5x+1). \]

Since powers, sums and products all obey the rules above, we may substitute \(x=2\) directly:

\[ \lim_{x\to 2}(3x^2-5x+1) = 3\cdot 2^2-5\cdot 2+1 = 12-10+1 = 3. \]

Consider now the limit

\[ \lim_{x\to 1}\frac{x^2+1}{x+2}. \]

The limit of the denominator is \(3\), and is therefore non-zero. We may thus apply the quotient rule:

\[ \lim_{x\to 1}\frac{x^2+1}{x+2} = \frac{1^2+1}{1+2} = \frac{2}{3}. \]

When the rules do not suffice

The rules above apply directly whenever the operations between the limits produce a determinate result. They cannot, however, be applied mechanically when expressions devoid of determinate meaning arise.

For instance, if

\[ \lim_{x\to x_0}f(x)=0 \qquad\text{and}\qquad \lim_{x\to x_0}g(x)=0, \]

we cannot conclude that

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)} \]

has a determinate value. The expression

\[ \frac{0}{0} \]

does not represent a result, but an indeterminate form.

In such cases the expression must be transformed, simplified, or treated with more specific tools. The principal indeterminate forms will be studied in the following section.

The same properties hold, with the appropriate modifications, for limits as \(x\to+\infty\), as \(x\to-\infty\), and for the right-hand and left-hand limits.

Indeterminate forms

In computing limits it may happen that the direct application of the rules on operations fails to determine the result. In such cases we speak of indeterminate forms.

An indeterminate form is not a number and is not a result. It is a situation in which knowledge of the individual limits is insufficient to establish the limit of the expression under consideration.

For instance, if

\[ \lim_{x\to x_0}f(x)=0 \qquad\text{and}\qquad \lim_{x\to x_0}g(x)=0, \]

we cannot deduce directly the limit of the quotient

\[ \frac{f(x)}{g(x)}. \]

Indeed, depending on the functions involved, the limit may be a real number, may be infinite, or may fail to exist.

The indeterminate form \(0/0\)

The form

\[ \frac{0}{0} \]

arises when both numerator and denominator tend to zero.

For example:

\[ \lim_{x\to 1}\frac{x^2-1}{x-1}. \]

Substituting \(x=1\) formally gives the form \(0/0\). Nevertheless, for \(x\neq 1\), we may simplify:

\[ \frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1. \]

Hence:

\[ \lim_{x\to 1}\frac{x^2-1}{x-1} = \lim_{x\to 1}(x+1) = 2. \]

This shows that the form \(0/0\) does not signify that the limit equals zero, but rather that the expression must be transformed.

The indeterminate form \(\infty/\infty\)

The form

\[ \frac{\infty}{\infty} \]

arises when both numerator and denominator become arbitrarily large in absolute value.

For example:

\[ \lim_{x\to+\infty}\frac{3x^2+1}{x^2-5}. \]

Both numerator and denominator tend to \(+\infty\). To evaluate the limit, we may divide numerator and denominator by \(x^2\):

\[ \frac{3x^2+1}{x^2-5} = \frac{3+\displaystyle\frac{1}{x^2}}{1-\displaystyle\frac{5}{x^2}}. \]

Since \(\displaystyle\frac{1}{x^2}\to 0\) and \(\displaystyle\frac{5}{x^2}\to 0\) as \(x\to+\infty\), we obtain:

\[ \lim_{x\to+\infty}\frac{3x^2+1}{x^2-5}=3. \]

Here too the notation \(\infty/\infty\) represents no result in itself: it merely signals that the expression requires closer analysis.

The indeterminate form \(\infty-\infty\)

The form

\[ \infty-\infty \]

occurs when two divergent quantities are subtracted from one another. The outcome depends on how quickly the two quantities grow.

For example:

\[ \lim_{x\to+\infty}\left(\sqrt{x^2+x}-x\right). \]

Both terms tend to \(+\infty\), so a form \(\infty-\infty\) arises. To resolve it, we rationalise:

\[ \sqrt{x^2+x}-x = \frac{(\sqrt{x^2+x}-x)(\sqrt{x^2+x}+x)}{\sqrt{x^2+x}+x} = \frac{x}{\sqrt{x^2+x}+x}. \]

As \(x\to+\infty\), we may divide numerator and denominator by \(x\):

\[ \frac{x}{\sqrt{x^2+x}+x} = \frac{1}{\sqrt{1+\displaystyle\frac{1}{x}}+1}. \]

Hence:

\[ \lim_{x\to+\infty}\left(\sqrt{x^2+x}-x\right) = \frac{1}{2}. \]

The indeterminate form \(0\cdot\infty\)

The form

\[ 0\cdot\infty \]

arises when one factor tends to zero while the other becomes arbitrarily large in absolute value.

In such cases one often seeks to recast the product as a quotient, so as to reduce the problem to a form \(0/0\) or \(\infty/\infty\).

For example:

\[ \lim_{x\to 0^+}x\ln x. \]

As \(x\to 0^+\), we have \(x\to 0\) and \(\ln x\to-\infty\), so a form \(0\cdot(-\infty)\) arises. We may rewrite:

\[ x\ln x=\frac{\ln x}{\displaystyle\frac{1}{x}}. \]

In this way the limit is reduced to a form \(\infty/\infty\), which can be treated with suitable tools. In particular, one obtains:

\[ \lim_{x\to 0^+}x\ln x=0. \]

Indeterminate exponential forms

There also exist indeterminate forms involving powers whose base and exponent both vary. The principal ones are:

\[ 1^\infty, \qquad 0^0, \qquad \infty^0. \]

These forms arise in the study of limits of the type

\[ \lim_{x\to x_0}\bigl(f(x)\bigr)^{g(x)}, \]

when the base \(f(x)\) and the exponent \(g(x)\) vary simultaneously. In such cases one requires, at least throughout a deleted neighbourhood of the point considered, that the base be positive, so as to make use of the exponential representation

\[ \bigl(f(x)\bigr)^{g(x)} = e^{g(x)\ln(f(x))}. \]

The study of the limit is thereby reduced to computing the limit of the exponent \(g(x)\ln(f(x))\).

List of the principal indeterminate forms

The principal indeterminate forms are:

\[ \frac{0}{0}, \qquad \frac{\infty}{\infty}, \qquad \infty-\infty, \qquad 0\cdot\infty, \qquad 1^\infty, \qquad 0^0, \qquad \infty^0. \]

When an indeterminate form arises, one must never assign a value to the limit automatically. Instead, the expression must be transformed, standard limits invoked, comparisons applied, or other tools of analysis brought to bear.

Forms that are not indeterminate

Not every expression involving zero or infinity is indeterminate. For instance, if \(L\in\mathbb{R}\), then in many cases the ratio of a quantity tending to \(L\) to a quantity tending to infinity tends to zero:

\[ \frac{L}{\infty}=0. \]

This notation is merely an intuitive shorthand: its rigorous meaning is that the numerator tends to a finite real number while the denominator becomes arbitrarily large in absolute value.

Similarly, expressions such as \(L+\infty\), with \(L\in\mathbb{R}\), are not indeterminate forms: the infinite term dominates the finite one.

The distinction between determinate and indeterminate forms is essential, since it tells us when the rules on limits yield an immediate answer and when, instead, further work is required.

Standard limits

The standard limits are fundamental limits that recur frequently in the study of functions. They allow many indeterminate forms, particularly those of type \(0/0\), to be resolved by reducing the expression to limits already known.

These limits must not be applied mechanically: one must always verify that the variable or expression considered tends to the value required, and that the functions involved are defined throughout a deleted neighbourhood of the point in question.

The standard limit of sine

One of the most important standard limits is

\[ \lim_{x\to 0}\frac{\sin x}{x}=1. \]

This limit holds when the angle \(x\) is measured in radians. It states that, for values of \(x\) close to \(0\), the sine of \(x\) behaves like \(x\) itself.

Equivalently, as \(x\to 0\) we may write informally:

\[ \sin x \sim x. \]

The notation \(\sin x \sim x\) means that the ratio of \(\sin x\) to \(x\) tends to \(1\).

The standard limit of cosine

Another fundamental limit is

\[ \lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}. \]

It describes the behaviour of \(1-\cos x\) near \(0\). In particular, as \(x\to 0\),

\[ 1-\cos x \sim \frac{x^2}{2}. \]

This limit is often useful whenever trigonometric expressions arise in indeterminate form.

The standard limit of tangent

From the standard limit of sine, together with the continuity of cosine at \(0\), we obtain:

\[ \lim_{x\to 0}\frac{\tan x}{x}=1. \]

Indeed,

\[ \frac{\tan x}{x} = \frac{\sin x}{x}\cdot\frac{1}{\cos x}. \]

Since \(\displaystyle\frac{\sin x}{x}\to 1\) and \(\cos x\to 1\), it follows that \(\displaystyle\frac{\tan x}{x}\to 1\).

The standard exponential limit

A fundamental limit connected with Euler's number \(e\) is

\[ \lim_{x\to 0}(1+x)^{\frac{1}{x}}=e. \]

Here \(x\) is taken sufficiently close to \(0\), with \(x\neq 0\), and such that \(1+x>0\).

An equivalent form of the same limit is

\[ \lim_{x\to+\infty}\left(1+\frac{1}{x}\right)^x=e. \]

These limits underlie many transformations involving expressions of the type \(1^\infty\).

The standard limit of the logarithm

For the natural logarithm the following standard limit holds:

\[ \lim_{x\to 0}\frac{\ln(1+x)}{x}=1. \]

The function is defined for \(1+x>0\), that is, for \(x>-1\). The limit states that, as \(x\to 0\), the logarithm \(\ln(1+x)\) behaves like \(x\):

\[ \ln(1+x)\sim x. \]

More generally, if \(u(x)\to 0\) and \(1+u(x)>0\) throughout a deleted neighbourhood of the point considered, then

\[ \frac{\ln(1+u(x))}{u(x)}\to 1 \]

in the same limit process.

The standard limit of the exponential function

For the natural exponential function we have

\[ \lim_{x\to 0}\frac{e^x-1}{x}=1. \]

This limit states that, near \(0\), the quantity \(e^x-1\) behaves like \(x\):

\[ e^x-1\sim x. \]

More generally, if \(a>0\), then

\[ \lim_{x\to 0}\frac{a^x-1}{x}=\ln a. \]

Indeed \(a^x=e^{x\ln a}\), so the behaviour of \(a^x-1\) near \(0\) depends on the factor \(\ln a\).

The standard limit of powers

If \(\alpha\in\mathbb{R}\), the following limit holds:

\[ \lim_{x\to 0}\frac{(1+x)^\alpha-1}{x}=\alpha. \]

Here too one must consider \(x\) within a neighbourhood of \(0\) in which the real power \((1+x)^\alpha\) is defined. In particular, it suffices to require \(1+x>0\).

This limit is very useful whenever roots or powers with real exponents arise. For example, taking \(\alpha=\displaystyle\frac{1}{2}\), we obtain:

\[ \lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}=\frac{1}{2}. \]

A worked example

Let us compute the limit

\[ \lim_{x\to 0}\frac{\sin(3x)}{x}. \]

Multiply and divide by \(3\):

\[ \frac{\sin(3x)}{x} = 3\cdot\frac{\sin(3x)}{3x}. \]

Since \(3x\to 0\) as \(x\to 0\), the standard limit of sine gives

\[ \frac{\sin(3x)}{3x}\to 1. \]

Hence:

\[ \lim_{x\to 0}\frac{\sin(3x)}{x}=3. \]

Table of the principal standard limits

We summarise the principal standard limits below:

\[ \lim_{x\to 0}\frac{\sin x}{x}=1, \qquad \lim_{x\to 0}\frac{\tan x}{x}=1, \qquad \lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}. \]

\[ \lim_{x\to 0}\frac{\ln(1+x)}{x}=1, \qquad \lim_{x\to 0}\frac{e^x-1}{x}=1, \qquad \lim_{x\to 0}\frac{a^x-1}{x}=\ln a. \]

\[ \lim_{x\to 0}(1+x)^{\frac{1}{x}}=e, \qquad \lim_{x\to+\infty}\left(1+\frac{1}{x}\right)^x=e, \qquad \lim_{x\to 0}\frac{(1+x)^\alpha-1}{x}=\alpha. \]

The equivalent forms derived from these limits are often decisive in the computation of limits. They must, however, be used only when the argument genuinely tends to \(0\), or when the variable tends to infinity in the manner required by the formula.

Infinitesimals and infinities

In the study of limits it is often useful to describe a function not merely by the value of its limit, but also by the rate at which it tends to zero or becomes arbitrarily large.

This need leads to the notions of infinitesimal, infinity, and comparison of orders.

Infinitesimals

A function \(f\) is said to be infinitesimal as \(x\to x_0\) if

\[ \lim_{x\to x_0}f(x)=0. \]

In other words, an infinitesimal is a function that, in the limit process considered, takes values arbitrarily close to zero.

For instance, as \(x\to 0\), the following functions are infinitesimal:

\[ x, \qquad x^2, \qquad \sin x, \qquad 1-\cos x. \]

Indeed, all of these functions tend to zero as \(x\to 0\).

The function \(\displaystyle\frac{1}{x}\) is likewise infinitesimal, but as \(x\to+\infty\) or as \(x\to-\infty\), since

\[ \lim_{x\to+\infty}\frac{1}{x}=0 \qquad\text{and}\qquad \lim_{x\to-\infty}\frac{1}{x}=0. \]

Infinities

A function \(f\) is said to be infinite as \(x\to x_0\) if

\[ \lim_{x\to x_0}f(x)=+\infty \]

or

\[ \lim_{x\to x_0}f(x)=-\infty. \]

More generally, a function is said to be infinite when the values of \(f(x)\) become arbitrarily large in absolute value in the limit process considered.

For instance, as \(x\to+\infty\), the following are infinite functions:

\[ x, \qquad x^2, \qquad e^x. \]

As \(x\to 0\), on the other hand, the function

\[ \frac{1}{x^2}, \]

is infinite, since

\[ \lim_{x\to 0}\frac{1}{x^2}=+\infty. \]

The relation between infinitesimals and infinities

The notions of infinitesimal and infinite are closely related. If \(f(x)\) is infinitesimal and \(f(x)\neq 0\) throughout a deleted neighbourhood of the point considered, then the reciprocal function

\[ \frac{1}{f(x)} \]

is infinite, except where the sign of \(f(x)\) produces different behaviour from the right and from the left.

For instance, as \(x\to 0\), the function \(x^2\) is infinitesimal and positive for \(x\neq 0\). Consequently,

\[ \frac{1}{x^2} \]

is positively infinite as \(x\to 0\).

Similarly, if \(f(x)\) is infinite and does not vanish throughout a deleted neighbourhood of the point considered, then

\[ \frac{1}{f(x)} \]

is infinitesimal.

Comparing infinitesimals

Two infinitesimals may tend to zero at different rates. To compare them, we study the limit of their ratio.

Let \(f\) and \(g\) be two infinitesimals as \(x\to x_0\), with \(g(x)\neq 0\) throughout a deleted neighbourhood of \(x_0\). Consider the limit

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}. \]

If

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}=0, \]

then \(f\) is an infinitesimal of higher order than \(g\). This means that \(f(x)\) tends to zero more rapidly than \(g(x)\).

If instead

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}=\ell, \qquad \ell\in\mathbb{R},\quad \ell\neq 0, \]

then \(f\) and \(g\) are infinitesimals of the same order.

If, finally,

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}=\pm\infty, \]

then \(f\) tends to zero more slowly than \(g\).

An example on comparing infinitesimals

As \(x\to 0\), let us compare the infinitesimals \(x^2\) and \(x\). We compute:

\[ \lim_{x\to 0}\frac{x^2}{x} = \lim_{x\to 0}x = 0. \]

Hence \(x^2\) is an infinitesimal of higher order than \(x\): indeed, \(x^2\) tends to zero more rapidly than \(x\).

Let us now compare \(\sin x\) and \(x\), again as \(x\to 0\). From the standard limit

\[ \lim_{x\to 0}\frac{\sin x}{x}=1 \]

it follows that \(\sin x\) and \(x\) are infinitesimals of the same order.

Equivalent infinities and equivalent infinitesimals

Two functions \(f\) and \(g\) are said to be equivalent as \(x\to x_0\) if

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}=1. \]

In that case we write

\[ f(x)\sim g(x) \qquad\text{as }x\to x_0. \]

The notation \(f(x)\sim g(x)\) means that, in the limit process considered, the two functions display the same principal behaviour.

For instance, as \(x\to 0\), the standard limits yield the equivalences

\[ \sin x\sim x, \qquad \tan x\sim x, \qquad 1-\cos x\sim \frac{x^2}{2}, \qquad \ln(1+x)\sim x, \qquad e^x-1\sim x. \]

These equivalences are very useful in the computation of limits, since they allow us to replace a function by a simpler one having the same principal behaviour.

Correct use of equivalents

Equivalents must be used with care. In particular, substitution by means of equivalents is safe within products and quotients, but cannot be applied mechanically within sums or differences, where cancellation of the leading terms may occur.

For example, since \(\sin x\sim x\) as \(x\to 0\), we may compute:

\[ \lim_{x\to 0}\frac{\sin x}{x}=1. \]

In an expression such as

\[ \sin x-x, \]

however, we cannot simply replace \(\sin x\) by \(x\) and conclude that the difference vanishes. In reality, the difference is of higher order and requires more refined tools, such as expansions or specific transformations.

This observation is fundamental: equivalents describe the principal behaviour of a function, but may not suffice when the leading terms cancel.

Comparing infinities

Infinite functions may likewise be compared by means of their ratio. If \(f\) and \(g\) are infinite as \(x\to x_0\), we study

\[ \lim_{x\to x_0}\frac{f(x)}{g(x)}. \]

If the limit is \(0\), then \(f\) grows more slowly than \(g\). If the limit is a non-zero real number, the two functions are infinities of the same order. If the limit is infinite, then \(f\) grows more rapidly than \(g\).

For example, as \(x\to+\infty\),

\[ \lim_{x\to+\infty}\frac{x}{x^2} = \lim_{x\to+\infty}\frac{1}{x} = 0. \]

Hence \(x\) is an infinity of lower order than \(x^2\), that is, \(x^2\) grows more rapidly than \(x\).

More generally, as \(x\to+\infty\), the positive powers of \(x\) grow more rapidly the larger the exponent.

Remarks

The comparison of infinitesimals and infinities does not concern only the value of the limit, but also the rate at which a function tends to zero or diverges. This point of view is essential in resolving many indeterminate forms.

In particular, many techniques for computing limits consist in identifying the dominant term, that is, the term that determines the principal behaviour of the expression in the limit process considered.

Strategies for computing limits

Computing a limit does not consist in always applying the same rule. Depending on the form of the expression, it may be necessary to use algebraic properties, standard limits, comparisons, equivalences, or specific transformations.

A sound strategy begins by recognising whether the expression leads to a determinate form or to an indeterminate one.

Direct substitution where possible

When the rules on limits apply directly and no indeterminate forms arise, the limit is computed by substituting the value to which \(x\) tends.

For example:

\[ \lim_{x\to 2}(x^2+3x-1) = 2^2+3\cdot 2-1 = 9. \]

Here no difficulty arises: polynomials, sums and products all behave regularly with respect to the limit.

Simplification in forms of type \(0/0\)

When a form \(0/0\) arises, one of the first strategies is to simplify the expression, if possible.

For example:

\[ \lim_{x\to 3}\frac{x^2-9}{x-3}. \]

Substituting \(x=3\) formally gives \(0/0\). We factorise the numerator:

\[ x^2-9=(x-3)(x+3). \]

For \(x\neq 3\), we may therefore write:

\[ \frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3. \]

It follows that

\[ \lim_{x\to 3}\frac{x^2-9}{x-3} = \lim_{x\to 3}(x+3) = 6. \]

This simplification is legitimate because the limit studies the behaviour for \(x\) close to \(3\), but distinct from \(3\).

Rationalisation

When radicals and differences appear, it can be useful to multiply and divide by the conjugate expression.

Consider:

\[ \lim_{x\to 0}\frac{\sqrt{1+x}-1}{x}. \]

Substituting \(x=0\) formally gives \(0/0\). We rationalise:

\[ \frac{\sqrt{1+x}-1}{x} = \frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{x(\sqrt{1+x}+1)} = \frac{1+x-1}{x(\sqrt{1+x}+1)}. \]

Since \(1+x-1=x\), for \(x\neq 0\) we obtain:

\[ \frac{\sqrt{1+x}-1}{x} = \frac{1}{\sqrt{1+x}+1}. \]

Hence:

\[ \lim_{x\to 0}\frac{\sqrt{1+x}-1}{x} = \frac{1}{2}. \]

Dividing by the dominant term

For limits of rational functions as \(x\to+\infty\) or as \(x\to-\infty\), a fundamental strategy is to divide numerator and denominator by the highest power present.

For example:

\[ \lim_{x\to+\infty}\frac{2x^3-x+1}{5x^3+4x^2-7}. \]

The dominant term is \(x^3\). We divide numerator and denominator by \(x^3\):

\[ \frac{2x^3-x+1}{5x^3+4x^2-7} = \frac{2-\displaystyle\frac{1}{x^2}+\displaystyle\frac{1}{x^3}}{5+\displaystyle\frac{4}{x}-\displaystyle\frac{7}{x^3}}. \]

Since

\[ \frac{1}{x}\to 0, \qquad \frac{1}{x^2}\to 0, \qquad \frac{1}{x^3}\to 0 \]

as \(x\to+\infty\), we obtain:

\[ \lim_{x\to+\infty}\frac{2x^3-x+1}{5x^3+4x^2-7} = \frac{2}{5}. \]

Using the standard limits

When trigonometric, logarithmic, exponential, or power functions appear, many limits can be reduced to the standard limits.

For example:

\[ \lim_{x\to 0}\frac{\ln(1+4x)}{x}. \]

Multiply and divide by \(4\):

\[ \frac{\ln(1+4x)}{x} = 4\cdot\frac{\ln(1+4x)}{4x}. \]

Since \(4x\to 0\) as \(x\to 0\), the standard limit

\[ \lim_{t\to 0}\frac{\ln(1+t)}{t}=1 \]

gives

\[ \lim_{x\to 0}\frac{\ln(1+4x)}{x}=4. \]

Using equivalent infinitesimals

Equivalent infinitesimals allow us to replace, within products and quotients, one function by another, simpler function having the same principal behaviour.

For example, as \(x\to 0\), we know that

\[ \sin x\sim x \qquad\text{and}\qquad e^x-1\sim x. \]

Hence:

\[ \lim_{x\to 0}\frac{\sin x}{e^x-1} = 1. \]

Indeed, numerator and denominator are both equivalent to \(x\).

This method is very swift, but must be used with care: equivalents are safe within products and quotients, while within sums and differences they can lead to errors if the leading terms cancel.

Using the squeeze theorem

When a function is difficult to handle directly, but can be trapped between two functions with the same limit, one may invoke the squeeze theorem.

For example:

\[ \lim_{x\to 0}x^2\cos\frac{1}{x}. \]

Since, for every \(x\neq 0\),

\[ -1\leq \cos\frac{1}{x}\leq 1, \]

multiplying by \(x^2\geq 0\) gives:

\[ -x^2\leq x^2\cos\frac{1}{x}\leq x^2. \]

Since

\[ \lim_{x\to 0}(-x^2)=0 \qquad\text{and}\qquad \lim_{x\to 0}x^2=0, \]

the squeeze theorem gives

\[ \lim_{x\to 0}x^2\cos\frac{1}{x}=0. \]

Studying the right- and left-hand limits separately

When the expression changes its behaviour according to the sign of \(x-x_0\), the right-hand and left-hand limits must be studied separately.

Consider, for instance,

\[ \lim_{x\to 0}\frac{|x|}{x}. \]

For \(x>0\), we have \(|x|=x\), so that

\[ \frac{|x|}{x}=1. \]

For \(x<0\), we have \(|x|=-x\), so that

\[ \frac{|x|}{x}=-1. \]

Hence:

\[ \lim_{x\to 0^+}\frac{|x|}{x}=1, \qquad \lim_{x\to 0^-}\frac{|x|}{x}=-1. \]

Since the right-hand and left-hand limits differ, the limit as \(x\to 0\) does not exist.

Identifying the dominant term

In many expressions, particularly as \(x\to+\infty\) or as \(x\to-\infty\), the behaviour of the limit is governed by the dominant term, that is, by the term that grows most rapidly.

For example:

\[ \lim_{x\to+\infty}(x^3-4x^2+7x). \]

The dominant term is \(x^3\). The remaining terms grow more slowly and do not alter the principal behaviour. Hence:

\[ \lim_{x\to+\infty}(x^3-4x^2+7x)=+\infty. \]

As \(x\to-\infty\), on the other hand, the dominant term is again \(x^3\), but here \(x^3\to-\infty\). Thus:

\[ \lim_{x\to-\infty}(x^3-4x^2+7x)=-\infty. \]

A working scheme

In summary, to compute a limit it is advisable to proceed as follows:

1. identify the point or the direction towards which the variable tends;
2. check whether the rules on limits apply directly;
3. identify any indeterminate forms;
4. choose a suitable transformation: factorisation, simplification, rationalisation, division by the dominant term, standard limits, equivalents, or comparison;
5. if necessary, study the right-hand and left-hand limits separately;
6. draw a conclusion only after verifying that the conditions used are valid in the limit process considered.

The essential point is not to confuse symbolic expressions with automatic results. An indeterminate form signals that the limit requires a more precise analysis; a determinate form, by contrast, often allows one to conclude directly by applying the properties of limits.

Graphical interpretation of limits and asymptotes

The concept of limit has a strong graphical interpretation. Studying a limit amounts to observing the behaviour of the graph of a function as the point with abscissa \(x\) approaches a fixed value, or as \(x\) recedes indefinitely towards \(+\infty\) or towards \(-\infty\).

The graph does not replace the rigorous definition, but helps to visualise the meaning of the various situations that may arise.

A finite limit at a point

If

\[ \lim_{x\to x_0}f(x)=L, \]

then, as \(x\) approaches \(x_0\), the points of the graph of \(f\) approach the height \(L\).

This does not necessarily mean that the graph passes through the point \((x_0,L)\). Indeed, the value \(f(x_0)\) may not be defined, or may differ from \(L\).

Graphically, then, the limit describes the trend of the graph near the vertical line \(x=x_0\), but does not necessarily depend on the point of the graph with abscissa \(x_0\).

Right-hand and left-hand limits in the graph

The right-hand limit describes the behaviour of the graph as we approach \(x_0\) from values greater than \(x_0\). The left-hand limit describes instead the behaviour of the graph as we approach \(x_0\) from values less than \(x_0\).

If

\[ \lim_{x\to x_0^-}f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x)=L, \]

then the graph approaches the same height \(L\) from both sides, and the limit as \(x\to x_0\) exists.

If instead the right-hand and left-hand limits differ, the graph approaches two different heights. In this case the limit as \(x\to x_0\) does not exist.

Infinite limits and vertical asymptotes

If, as \(x\) approaches \(x_0\), the values of the function become arbitrarily large or arbitrarily small, the graph approaches the vertical line \(x=x_0\).

If at least one of the right-hand and left-hand limits is infinite, then the line

\[ x=x_0 \]

is a vertical asymptote for the graph of the function.

For example, if

\[ \lim_{x\to x_0^+}f(x)=+\infty, \]

then, approaching \(x_0\) from the right, the graph rises without bound along the direction of the vertical line \(x=x_0\).

Similarly, if

\[ \lim_{x\to x_0^-}f(x)=-\infty, \]

then, approaching \(x_0\) from the left, the graph descends without bound along the same vertical line.

It is thus possible for the behaviour from the right and from the left to differ. For instance, a function may tend to \(+\infty\) on one side and to \(-\infty\) on the other.

A finite limit at infinity and horizontal asymptotes

If a function tends to a real number \(L\) as \(x\to+\infty\), then the graph approaches the horizontal line

\[ y=L \]

as we move indefinitely towards the right.

In this case the line \(y=L\) is a right-hand horizontal asymptote for the graph of the function.

Similarly, if

\[ \lim_{x\to-\infty}f(x)=M, \]

then the line

\[ y=M \]

is a left-hand horizontal asymptote.

The two horizontal asymptotes may coincide or may differ. For instance, a function may tend to one value as \(x\to+\infty\) and to a different value as \(x\to-\infty\).

Oblique asymptotes

Besides vertical and horizontal asymptotes, a function may possess an oblique asymptote (also called a slant asymptote). This occurs when, as \(x\to+\infty\) or as \(x\to-\infty\), the graph of the function approaches a non-horizontal line.

A line with equation

\[ y=mx+q, \qquad m\neq 0, \]

is an oblique asymptote for \(f\) as \(x\to+\infty\) if

\[ \lim_{x\to+\infty}\bigl(f(x)-(mx+q)\bigr)=0. \]

The same definition applies analogously as \(x\to-\infty\), with the limit process adjusted accordingly.

The condition above means that the vertical distance between the graph of the function and the line \(y=mx+q\) tends to zero.

When an oblique asymptote exists, the coefficients \(m\) and \(q\) are computed, in the ordinary case, by means of the limits

\[ m=\lim_{x\to+\infty}\frac{f(x)}{x} \]

and

\[ q=\lim_{x\to+\infty}\bigl(f(x)-mx\bigr), \]

provided these limits exist, are finite, and \(m\neq 0\). For \(x\to-\infty\) the same formulae are used with the limit taken as \(x\to-\infty\).

Examples of graphical interpretation

Consider the function

\[ f(x)=\frac{1}{x}. \]

As \(x\to 0^+\),

\[ \lim_{x\to 0^+}\frac{1}{x}=+\infty, \]

whereas as \(x\to 0^-\),

\[ \lim_{x\to 0^-}\frac{1}{x}=-\infty. \]

The line \(x=0\), that is, the \(y\)-axis, is thus a vertical asymptote for the graph of the function.

Moreover,

\[ \lim_{x\to+\infty}\frac{1}{x}=0 \qquad\text{and}\qquad \lim_{x\to-\infty}\frac{1}{x}=0. \]

Hence the line \(y=0\), that is, the \(x\)-axis, is both a right-hand and a left-hand horizontal asymptote.

Consider now the function

\[ g(x)=x+\frac{1}{x}. \]

As \(x\to+\infty\),

\[ g(x)-x=\frac{1}{x}\to 0. \]

Hence the line

\[ y=x \]

is an oblique asymptote as \(x\to+\infty\). The same holds as \(x\to-\infty\), since \(\displaystyle\frac{1}{x}\to 0\) in that direction as well.

Concluding remarks

The graphical interpretation of limits allows us to connect the rigorous definition with the visible behaviour of the graph. The graph should, however, be regarded as a guide, not as a proof.

To establish with certainty the existence and value of a limit, one must always refer to the definitions, theorems and properties studied in the preceding sections.

In summary, limits allow us to describe with precision three fundamental aspects of the behaviour of a function: what happens near a point, what happens at infinity, and how the graph is positioned with respect to its asymptotes.