Special Limits

Special limits are frequently encountered limits involving functions or sequences that are particularly useful for evaluating many other limits quickly and effectively.

Why memorize a special limit? Understanding the reasoning behind a special limit isn’t always straightforward - it often requires a formal proof. However, once you’ve memorized the result, it becomes a powerful shortcut for solving a wide range of limit problems efficiently.

List of Special Limits

Here’s a list of some of the most important special limits in calculus:

$$ \lim_{x \rightarrow \infty} \left(1+\frac{1}{x}\right)^x = e $$
$$ \lim_{x \rightarrow \infty} \left(1+\frac{k}{x}\right)^x = e^k $$
$$ \lim_{x \rightarrow 0} \frac{ \sin x }{x} = 1 $$
$$ \lim_{x \rightarrow 0} \frac{ 1 - \cos x }{x^2} = \frac{1}{2} $$
$$ \lim_{x \rightarrow 0} \frac{ 1 - \cos kx }{x^2} = \frac{k^2}{2} $$
$$ \lim_{x \rightarrow 0} \frac{ \log(1+x) }{x} = 1 $$
$$ \lim_{x \rightarrow \infty} a^x = \begin{cases} +\infty \quad \text{if } a > 1 \\ 1 \quad \text{if } a = 1 \\ 0 \quad \text{if } -1 < a < 1 \\ \text{does not exist} \quad \text{if } a \le -1 \\ \end{cases} $$	Proof
$$ \lim_{x \rightarrow \infty} \sqrt[x]{a} = 1 $$	Proof
$$ \lim_{x \rightarrow \infty} \sqrt[x]{x^b} = 1 $$
$$ \lim_{x \rightarrow \infty} \left(1+\frac{(1+x)^c - 1}{x}\right)^x = c $$
$$ \lim_{x \rightarrow 0} \frac{(1+x)^c - 1}{x} = c $$
$$ \lim_{x \rightarrow 0} \frac{ \tan x }{x} = 1 $$
$$ \lim_{x \rightarrow 0} \frac{ \arcsin x }{x} = 1 $$
$$ \lim_{x \rightarrow 0} \frac{ \arctan x }{x} = 1 $$
$$ \lim_{x \rightarrow 0} x \cdot \log x = 0 $$
$$ \lim_{x \rightarrow 0} \frac{ \log_b{(1+x)} }{x} = \frac{1}{ \log b } $$
$$ \lim_{x \rightarrow 0} \frac{ e^x - 1 }{ x } = 1 $$
$$ \lim_{x \rightarrow 0} \frac{ a^x - 1 }{ x } = \log a $$