Limits of functions

0/0 forms: factoring and rationalizing

If substituting $x = a$ gives $\frac{0}{0}$, the numerator and denominator share the factor $x - a$. Factor and cancel before substituting.

With square roots, rationalize:

Trigonometric limits

Create the form $\frac{\sin \theta}{\theta}$ first, as in $\frac{\sin 3x}{2x} = \frac{\sin 3x}{3x} \cdot \frac{3}{2}$. Multiply expressions with $1 - \cos x$ by $1 + \cos x$ to get $\sin^2 x$.

For limits such as $x \to \pi$, substitute $t = x - \pi$ to get a limit as $t \to 0$.

The number $e$ and exponential limits

Example: $\displaystyle\lim_{x \to 0} (1 + 2x)^{\frac{1}{x}} = \lim_{x \to 0} \left\{(1 + 2x)^{\frac{1}{2x}}\right\}^2 = e^2$. Adjust the exponent to match the definition. (Here $\log$ is the natural logarithm.)

One-sided limits and continuity

The right-hand limit $\displaystyle\lim_{x \to a^{+}} f(x)$ is taken as $x$ approaches $a$ from above, and the left-hand limit $\displaystyle\lim_{x \to a^{-}} f(x)$ from below. The limit $\displaystyle\lim_{x \to a} f(x)$ exists exactly when they agree.

$f$ is continuous at $x = a$ when $\displaystyle\lim_{x \to a} f(x) = f(a)$. Remove absolute values like $|x - a|$ by treating $x > a$ and $x < a$ separately.