Standard Limits

Certain limits recur so often — and cannot be found by mere substitution — that their results are memorised as standard limits. Each is stated below with the exact conditions under which it holds.

Algebraic standard limit. For any rational index $n$ and any $a$ for which the expressions are defined,

This follows from factoring $x^n - a^n = (x-a)(x^{n-1} + x^{n-2}a + \dots + a^{n-1})$ and cancelling $(x-a)$; the bracket has $n$ terms, each tending to $a^{n-1}$, giving $n\,a^{n-1}$.

Trigonometric standard limits (angle $x$ measured in radians):

Exponential and logarithmic standard limits:

The result $\lim_{x\to 0}\dfrac{e^x-1}{x}=1$ is just the case $a=e$ of the first (since $\log_e e = 1$).

How to use them — match the pattern. Each standard limit fits a shape. To apply one, algebraically force the expression into that exact shape, adjusting by a constant factor if the argument is a multiple of $x$. For example, $\dfrac{\sin 5x}{x} = 5\cdot\dfrac{\sin 5x}{5x}$, and as $x\to 0$ the factor $\dfrac{\sin 5x}{5x} \to 1$ (since $5x \to 0$), so the whole limit is $5\cdot 1 = 5$.