Limits

$\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$

Common limits #

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

Squeeze Theorem #

$f(x) \leq g(x) \leq h(x)$ for all $x$ near $c$, execpt possibly at $c$.

If $\lim_{x \to c} f(x) = \lim_{x \to c} h(x)$ = L, then $\lim_{x \to c} g(x) = L$

Asymptotes #

• Vertical asymptote $x = c$

$$\lim_{x \to c^\pm} f(x) = \pm \infty$$

• Horizontal asymptote $y = b$:

$$\lim_{x \to \pm\infty} f(x) = b$$

L'Hospitals Rule #

If limit is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ then:

$$\lim_{x \to u} \frac{f(x)}{g(x)} = \lim_{x \to u} \frac{f'(x)}{g'(x)}$$

Solving limits #

• Limit with product, write as fraction: $a \cdot b = \frac{a}{1/b}$
• Powers: use logorithm laws:
  1. Solve $\lim_{x \to a} \ln{y}$ using: $y=f(x)^{g(x)} \Rightarrow \ln{y} = g(x)\ln{f(x)}$
  2. Use $y = e^{\ln{y}}$: $\lim_{x \to a} y = \lim_{x \to a} e^{\ln(y)}$