Limits and Continuity

Limits and Continuity

Definition

A function $f(x)$ has limit $L$ as $x$ approaches $a$ if $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $a$.

Key limit

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

Limit laws

• $\lim (f + g) = \lim f + \lim g$
• $\lim (f \cdot g) = \lim f \cdot \lim g$
• $\lim \frac{f}{g} = \frac{\lim f}{\lim g}$ (if $\lim g \neq 0$)

Continuity

$f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$.