In the study of the limit of a function it is not always sufficient to observe what happens as \(x\) approaches a point \(x_0\) without distinguishing the direction of approach. In many cases, indeed, the behaviour of the function may differ according to whether \(x\) tends to \(x_0\) through values greater than \(x_0\) or through values less than \(x_0\).
For this reason one introduces the right-hand limit and the left-hand limit. The right-hand limit describes the behaviour of a function as \(x\) approaches \(x_0\) while remaining greater than \(x_0\); the left-hand limit describes instead the behaviour of the function as \(x\) approaches \(x_0\) while remaining less than \(x_0\).
These two notions are fundamental in the study of piecewise-defined functions, of points of discontinuity, and of limits at the endpoints of an interval. They also allow one to state precisely when the limit as \(x\to x_0\), understood without any one-sided restriction, exists: this occurs exactly when the right-hand limit and the left-hand limit both exist and coincide.
Contents
- Intuitive idea of the right-hand and left-hand limit
- Approaching a point from the right and from the left
- Definition of the right-hand limit
- Definition of the left-hand limit
- Relation to the two-sided limit
- Right-hand and left-hand limits that differ
- Infinite right-hand and left-hand limits
- Examples with piecewise-defined functions
- Right-continuity and left-continuity
- Common mistakes to avoid
Intuitive idea of the right-hand and left-hand limit
When we study the limit of a function as \(x\to x_0\), we observe the behaviour of the values \(f(x)\) as \(x\) approaches the point \(x_0\). This approach to \(x_0\), however, can take place in two distinct ways: through values greater than \(x_0\), or through values less than \(x_0\).
If \(x\) approaches \(x_0\) while remaining greater than \(x_0\), we say that \(x\) tends to \(x_0\) from the right, and we write
\[ x\to x_0^+. \]
If instead \(x\) approaches \(x_0\) while remaining less than \(x_0\), we say that \(x\) tends to \(x_0\) from the left, and we write
\[ x\to x_0^-. \]
The right-hand limit thus describes the behaviour of the function as \(x\) approaches \(x_0\) through values \(x>x_0\). The left-hand limit describes instead the behaviour of the function as \(x\) approaches \(x_0\) through values \(x<x_0\).
This distinction matters because a function may behave differently to the right and to the left of the same point. In particular, it can happen that the values of \(f(x)\) approach one number on one side and a different number on the other.
The value taken by the function at the point \(x_0\), when it exists, is not what determines the limit. Even in the case of right-hand and left-hand limits, what matters is the behaviour of \(f(x)\) for values of \(x\) close to \(x_0\), but distinct from \(x_0\).
Approaching a point from the right and from the left
Let \(f:D\to\mathbb{R}\) be a real-valued function of a real variable, with domain \(D\subseteq\mathbb{R}\). To speak of the behaviour of \(f(x)\) as \(x\) approaches a point \(x_0\), it is not necessary that \(x_0\) belong to the domain of the function.
What matters is that there exist points of the domain arbitrarily close to \(x_0\). In the case of the right-hand limit, these points must lie to the right of \(x_0\); in the case of the left-hand limit, they must lie to the left of \(x_0\).
To say that \(x\) approaches \(x_0\) from the right means considering values of \(x\) belonging to the domain of the function and such that
\[ x_0<x<x_0+\delta \]
for smaller and smaller positive values of \(\delta\). In symbols, one writes
\[ x\to x_0^+. \]
To say that \(x\) approaches \(x_0\) from the left means instead considering values of \(x\) belonging to the domain of the function and such that
\[ x_0-\delta<x<x_0. \]
In this case one writes
\[ x\to x_0^-. \]
Thus, when studying a right-hand or left-hand limit, one does not necessarily observe every value of \(x\) close to \(x_0\), but only those that belong to the domain of the function and lie on the side under consideration.
For instance, if a function is defined on an interval of the form \([a,b]\), at the point \(a\) one can study the right-hand limit, since there exist points of the domain to the right of \(a\), but one cannot study a left-hand limit within that domain. Likewise, at the point \(b\) one can study the left-hand limit, but not the right-hand limit.
More precisely, the right-hand limit at \(x_0\) makes sense whenever there exist points of the domain arbitrarily close to \(x_0\) and greater than \(x_0\). The left-hand limit at \(x_0\) makes sense whenever there exist points of the domain arbitrarily close to \(x_0\) and less than \(x_0\).
Definition of the right-hand limit
Let \(f:D\to\mathbb{R}\) be a real-valued function of a real variable, with \(D\subseteq\mathbb{R}\), and let \(x_0\) be a point such that there exist points of the domain arbitrarily close to \(x_0\) and greater than \(x_0\).
To say that \(x_0\) has points of the domain arbitrarily close on the right means that, for every \(\delta>0\), there exists at least one point \(x\in D\) such that
\[ x_0<x<x_0+\delta. \]
Under these conditions, we say that the function \(f\) has right-hand limit equal to \(L\) as \(x\to x_0\), and we write
\[ \lim_{x\to x_0^+} f(x)=L, \]
if for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0<x<x_0+\delta \implies |f(x)-L|<\varepsilon. \]
In other words, the values \(f(x)\) can be made arbitrarily close to \(L\), provided that \(x\) is sufficiently close to \(x_0\), belongs to the domain of the function, and lies to the right of \(x_0\).
The condition \(x_0<x\) is essential: in the right-hand limit one does not observe the behaviour of the function for values of \(x\) less than \(x_0\). Moreover, just as in the case of the limit as \(x\to x_0\), the value of the function at \(x_0\) is of no consequence, even when \(x_0\in D\).
The right-hand limit depends solely on the behaviour of the function at points of the domain that lie to the right of \(x_0\) and that approach \(x_0\) indefinitely closely.
Definition of the left-hand limit
Let \(f:D\to\mathbb{R}\) be a real-valued function of a real variable, with \(D\subseteq\mathbb{R}\), and let \(x_0\) be a point such that there exist points of the domain arbitrarily close to \(x_0\) and less than \(x_0\).
To say that \(x_0\) has points of the domain arbitrarily close on the left means that, for every \(\delta>0\), there exists at least one point \(x\in D\) such that
\[ x_0-\delta<x<x_0. \]
Under these conditions, we say that the function \(f\) has left-hand limit equal to \(L\) as \(x\to x_0\), and we write
\[ \lim_{x\to x_0^-} f(x)=L, \]
if for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0-\delta<x<x_0 \implies |f(x)-L|<\varepsilon. \]
In other words, the values \(f(x)\) can be made arbitrarily close to \(L\), provided that \(x\) is sufficiently close to \(x_0\), belongs to the domain of the function, and lies to the left of \(x_0\).
The condition \(x<x_0\) is essential: in the left-hand limit one does not consider the behaviour of the function for values of \(x\) greater than \(x_0\). Here too, whatever value the function may take at the point \(x_0\) has no bearing on the limit.
The left-hand limit depends solely on the behaviour of the function at points of the domain that lie to the left of \(x_0\) and that approach \(x_0\) indefinitely closely.
Relation to the two-sided limit
The limit as \(x\to x_0\), understood without any one-sided restriction, requires the function to approach the same value regardless of the manner in which \(x\) tends to \(x_0\) within the domain.
In particular, if the domain of the function has points arbitrarily close to \(x_0\) both on the left and on the right, then the limit
\[ \lim_{x\to x_0}f(x) \]
exists and equals \(L\) if and only if both limits
\[ \lim_{x\to x_0^-}f(x) \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x) \]
exist and are equal to \(L\). In symbols:
\[ \lim_{x\to x_0}f(x)=L \iff \lim_{x\to x_0^-}f(x)=L \ \text{and}\ \lim_{x\to x_0^+}f(x)=L. \]
This equivalence shows that the limit as \(x\to x_0\), taken without one-sided restriction, is more demanding than the two one-sided limits taken separately. It is not enough for the function to behave regularly on just one side: the behaviour must be the same on both sides.
If instead the two limits exist but are different, then the limit as \(x\to x_0\), without one-sided restriction, does not exist. The function approaches two different values according to the direction from which \(x\) tends to \(x_0\).
At the endpoints of an interval, the situation differs. For instance, if a function is defined on \([a,b]\), at the point \(a\) one naturally studies the right-hand limit, while at the point \(b\) one naturally studies the left-hand limit. In these cases no check is required on both sides, because the domain itself is present on only one side.
Right-hand and left-hand limits that differ
It can happen that a function has a well-defined behaviour both to the left and to the right of a point \(x_0\), but that the two behaviours lead to different values.
Suppose, for instance, that
\[ \lim_{x\to x_0^-}f(x)=L_1 \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x)=L_2, \]
with \(L_1\neq L_2\). In this case the limit as \(x\to x_0\), taken without one-sided restriction, does not exist.
Indeed, approaching \(x_0\) from the left, the values of the function approach \(L_1\); approaching instead from the right, they approach \(L_2\). If \(L_1\) and \(L_2\) differ, there is no single value to which the function tends as \(x\) approaches \(x_0\).
This situation arises frequently with piecewise-defined functions. Consider, for example, the function
\[ f(x)= \begin{cases} 1, & x<0,\\ 2, & x>0. \end{cases} \]
As \(x\to 0^-\), the values of the function equal \(1\), so
\[ \lim_{x\to 0^-}f(x)=1. \]
As \(x\to 0^+\), instead, the values of the function equal \(2\), so
\[ \lim_{x\to 0^+}f(x)=2. \]
Since the two limits differ, the limit as \(x\to 0\), without one-sided restriction, does not exist:
\[ \lim_{x\to 0}f(x) \quad\text{does not exist.} \]
The essential point is that the separate existence of the left-hand limit and the right-hand limit is not enough. For the limit as \(x\to x_0\) to exist, the two values must coincide.
Infinite right-hand and left-hand limits
The right-hand limit and the left-hand limit need not be finite real numbers. It can happen that, as one approaches a point \(x_0\) from just one side, the values of the function grow without bound or decrease without bound.
Let \(f:D\to\mathbb{R}\) be a real-valued function of a real variable, with \(D\subseteq\mathbb{R}\), and suppose that the right-hand limit at \(x_0\) makes sense. Writing
\[ \lim_{x\to x_0^+}f(x)=+\infty \]
means that for every \(M>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0<x<x_0+\delta \implies f(x)>M. \]
In other words, the values of \(f(x)\) become greater than any prescribed positive real number, provided that \(x\) is sufficiently close to \(x_0\) from the right.
Likewise, writing
\[ \lim_{x\to x_0^+}f(x)=-\infty \]
means that for every \(M>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0<x<x_0+\delta \implies f(x)<-M. \]
In this case, approaching \(x_0\) from the right, the values of the function become less than any prescribed negative real number, however large in absolute value.
The definitions on the left are entirely analogous. If the left-hand limit at \(x_0\) makes sense, writing
\[ \lim_{x\to x_0^-}f(x)=+\infty \]
means that for every \(M>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0-\delta<x<x_0 \implies f(x)>M. \]
Writing instead
\[ \lim_{x\to x_0^-}f(x)=-\infty \]
means that for every \(M>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0-\delta<x<x_0 \implies f(x)<-M. \]
Consider, for example, the function
\[ f(x)=\frac{1}{x}. \]
As \(x\to 0^+\), the denominator is positive and ever closer to zero; consequently the values of the function are positive and grow without bound:
\[ \lim_{x\to 0^+}\frac{1}{x}=+\infty. \]
As \(x\to 0^-\), instead, the denominator is negative and ever closer to zero; the values of the function are negative and decrease without bound:
\[ \lim_{x\to 0^-}\frac{1}{x}=-\infty. \]
Here again the two behaviours do not agree. Hence the limit as \(x\to 0\), taken without one-sided restriction, is neither \(+\infty\) nor \(-\infty\).
More generally, if on one side the function tends to \(+\infty\) and on the other to \(-\infty\), the limit as \(x\to x_0\) does not exist as a single limit. The right-hand and left-hand limits both exist, but they describe incompatible behaviours.
Examples with piecewise-defined functions
Piecewise-defined functions are one of the settings in which the right-hand and left-hand limits prove most useful. In these cases, indeed, the expression defining the function may change according to the interval in which \(x\) lies.
When computing the limit at a point where the definition of the function changes, one must use the expression valid to the left of the point for the left-hand limit, and the expression valid to the right of the point for the right-hand limit.
Consider the function
\[ f(x)= \begin{cases} x+1, & x<1,\\ 5, & x=1,\\ 3-x, & x>1. \end{cases} \]
To compute the left-hand limit at \(x_0=1\), we must use the branch valid for \(x<1\), namely \(f(x)=x+1\). Hence
\[ \lim_{x\to 1^-}f(x)=\lim_{x\to 1^-}(x+1)=2. \]
For the right-hand limit, instead, we must use the branch valid for \(x>1\), namely \(f(x)=3-x\). Therefore
\[ \lim_{x\to 1^+}f(x)=\lim_{x\to 1^+}(3-x)=2. \]
The two limits coincide. Consequently the limit as \(x\to 1\), taken without one-sided restriction, exists and equals
\[ \lim_{x\to 1}f(x)=2. \]
Yet \(f(1)=5\). This shows once again that the value of the function at the point does not determine the limit: the limit depends on the values taken by the function near the point, not necessarily on the value taken at the point itself.
Consider now a second example:
\[ g(x)= \begin{cases} x^2, & x<2,\\ x+1, & x\ge 2. \end{cases} \]
To the left of \(2\), the function is given by \(g(x)=x^2\). Hence
\[ \lim_{x\to 2^-}g(x)=\lim_{x\to 2^-}x^2=4. \]
To the right of \(2\), including the point \(2\) itself, the function is given by \(g(x)=x+1\). For the right-hand limit, however, we consider values \(x>2\), so
\[ \lim_{x\to 2^+}g(x)=\lim_{x\to 2^+}(x+1)=3. \]
Since the two limits differ, the limit as \(x\to 2\), without one-sided restriction, does not exist.
In summary, for piecewise-defined functions the correct procedure consists in reading carefully the domain of each branch and in computing separately the behaviour of the function from the left and from the right.
Right-continuity and left-continuity
The right-hand and left-hand limits also allow one to define the continuity of a function from just one side. This is particularly useful at the endpoints of an interval and at points where a function is defined piecewise.
Let \(f:D\to\mathbb{R}\) be a real-valued function of a real variable and let \(x_0\in D\). Suppose that the right-hand limit of \(f\) at \(x_0\) makes sense. We say that \(f\) is continuous from the right at \(x_0\) if
\[ \lim_{x\to x_0^+}f(x)=f(x_0). \]
Equivalently, \(f\) is continuous from the right at \(x_0\) if for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0\le x<x_0+\delta \implies |f(x)-f(x_0)|<\varepsilon. \]
Likewise, suppose that the left-hand limit of \(f\) at \(x_0\) makes sense. We say that \(f\) is continuous from the left at \(x_0\) if
\[ \lim_{x\to x_0^-}f(x)=f(x_0). \]
In \(\varepsilon\)-\(\delta\) form, this means that for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, for every \(x\in D\),
\[ x_0-\delta<x\le x_0 \implies |f(x)-f(x_0)|<\varepsilon. \]
The difference from the mere right-hand or left-hand limit is important. In the limit one observes only the behaviour of the function near \(x_0\), without the value \(f(x_0)\) playing any role. In continuity, by contrast, the value of the function at the point must coincide with the value towards which the function tends.
For instance, if a function is defined on an interval \([a,b]\), continuity at \(a\), relative to the domain, is established through continuity from the right, since the domain contains no points to the left of \(a\). In this case the natural condition is
\[ \lim_{x\to a^+}f(x)=f(a). \]
Likewise, at the point \(b\) one considers continuity from the left:
\[ \lim_{x\to b^-}f(x)=f(b). \]
If instead \(x_0\) is an interior point of the domain, and the function is defined on both sides of \(x_0\), then continuity at \(x_0\) requires continuity from both the left and the right. In symbols:
\[ f \text{ is continuous at } x_0 \iff \lim_{x\to x_0^-}f(x)=f(x_0) \ \text{and}\ \lim_{x\to x_0^+}f(x)=f(x_0). \]
Common mistakes to avoid
The first mistake to avoid is confusing the right-hand or left-hand limit with the value of the function at the point. The limit describes the behaviour of \(f(x)\) as \(x\) approaches \(x_0\) from a given side; the value \(f(x_0)\), when it exists, concerns instead the function precisely at the point \(x_0\).
For example, if
\[ \lim_{x\to x_0^+}f(x)=L, \]
it does not follow that \(f(x_0)=L\). The right-hand limit depends on the values of the function for \(x>x_0\) near \(x_0\), not on the value taken at the point.
A second mistake consists in overlooking the domain of the function. When computing a right-hand limit, one must consider only the values \(x\in D\) such that
\[ x_0<x<x_0+\delta. \]
When computing a left-hand limit, one must instead consider only the values \(x\in D\) such that
\[ x_0-\delta<x<x_0. \]
It is not enough, then, merely to look at the position of \(x\) relative to \(x_0\): one must also check that those values actually belong to the domain of the function.
A third common mistake concerns piecewise-defined functions. At a point where the definition of the function changes, the left-hand limit must be computed using the branch valid to the left of the point, while the right-hand limit must be computed using the branch valid to the right of the point. Whatever value may be assigned to the function at that point must not be used to compute the limits from the right and from the left.
A fourth mistake consists in concluding that the limit as \(x\to x_0\), without one-sided restriction, exists merely because one of the two one-sided limits exists. This is not sufficient. When the domain has points arbitrarily close to \(x_0\) both on the left and on the right, the limit as \(x\to x_0\) exists only if the left-hand limit and the right-hand limit both exist and coincide.
In symbols, when both sides are present in the domain, the correct condition is
\[ \lim_{x\to x_0^-}f(x)=L \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x)=L. \]
Only in this case can one write
\[ \lim_{x\to x_0}f(x)=L. \]
Finally, one must draw a precise distinction between limit and continuity. The existence of the right-hand or left-hand limit does not, by itself, imply continuity from that side. To have continuity from the right at \(x_0\), for instance, it is not enough that the right-hand limit exist: it must also be equal to the value of the function at the point, that is
\[ \lim_{x\to x_0^+}f(x)=f(x_0). \]
Likewise, continuity from the left requires
\[ \lim_{x\to x_0^-}f(x)=f(x_0). \]
Keeping these aspects distinct — the side of approach, the domain, the value of the function at the point, and the coincidence of the two limits — allows one to treat right-hand and left-hand limits correctly and without ambiguity.