AP Calculus AB/BC - Unit 1: Limits and Continuity

Limit Rules:

• Limit of a Constant: $\lim_{{x \to a}} c = c$
• Basic Limit: $\lim_{{x \to a}} x = a$
• Squeeze Theorem: Let $f, g,$ and $h$ be functions such that for all $x \in [a, b]$: $$f(x) \leq h(x) \leq g(x)$$ If $$\lim_{{x \to c}} f(x) = \lim_{{x \to c}} g(x) = L$$ then $$\lim_{{x \to c}} h(x) = L$$
• L'Hôpital's Rule: If $$\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}$$ then $$\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}$$

Horizontal Asymptotes:

1. If the largest exponent in the numerator is < largest exponent in the denominator, then: $$\lim_{{x \to \pm\infty}} f(x) = 0$$
2. If the largest exponent in the numerator is > largest exponent in the denominator, then: $$\lim_{{x \to \pm\infty}} f(x) = \text{DNE}$$
3. If the largest exponent in the numerator = largest exponent in the denominator, then the quotient of the leading coefficients is the asymptote: $$\lim_{{x \to \pm\infty}} f(x) = \frac{a}{b}$$

Vertical Asymptotes (VA):

$f(x) = \frac{P(x)}{q(x)}$