| Notable Limits | Proof | |
|---|---|---|
| \[\lim_{x \to 0} \frac{\sin(x)}{x} = 1 \] | Proof | |
| \[\lim_{x \to 0} \frac{\tan(x)}{x} = 1\] | Proof | |
| \[\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}=\frac{1}{2}\] | Proof | |
| \[\lim_{x \to 0} \frac{\ln(1 + x)}{x}=1\] | Proof | |
| \[\lim_{x \to 0} \frac{e^x - 1}{x}=1\] | Proof | |
| \[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}=\frac{1}{2}\] | Proof | |
| \[\lim_{x \to \infty} \frac{1}{x^n}=0\] | Proof | |
| \[\lim_{x \to \infty} \frac{e^x}{x^n}=\infty\] | Proof | |
| \[\lim_{x \to \infty} \frac{x^n}{e^x}=0\] | Proof | |
| \[\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x=e\] | Proof | |
| \[\lim_{x \to \infty} \left( 1 + \frac{a}{x} \right)^x=e^a\] | Proof | |
| \[\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}\] | Proof | |
| \[\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e\] | Proof | |
| \[\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)\] | Proof | |
| \[\lim_{x \to 0} \frac{(1+x)^{\alpha} - 1}{x} = \alpha\] | Proof | |
| \[\lim_{x \to \infty} x\sin\left(\frac{1}{x}\right) = 1\] | Proof | |
| \[\lim_{x \to 0} \frac{e^x - e^{-x}}{2x} = 1\] | Proof | |
| \[\lim_{x \to 0} \frac{\sinh(x)}{x} = 1\] | Proof | |
| \[\lim_{x \to 0} \frac{\cosh(x)}{x} = 1\] | Proof | |
| \[\lim_{x \to 0} \frac{\tanh(x)}{x} = 1\] | Proof |
