Limits and Continuity

$\lim_{x \to a} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$.

Evaluating Limits

1. Direct substitution: try plugging in $x = a$.
2. Factoring: if $\frac{0}{0}$, factor and cancel.
3. Conjugate: for radical expressions.
4. L'Hôpital's Rule (HL): if $\frac{0}{0}$ or $\frac{\infty}{\infty}$, take derivatives of top and bottom.

Example

$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x+2)(x-2)}{x-2} = \lim_{x \to 2} (x+2) = 4$.

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