Squeeze Theorem

The Squeeze Theorem is a method for computing limits without excessive applications of limit laws.

DEFN: \(If f(x) \leq g(x) \leq h(x)\) when \(x\) is near \(a\) and lim_{x→ a} f(x) = lim_{x→ a} h(x) = L$ then \(\lim_{x\rightarrow a} g(x) = L\).

EX:

\(\frac{\sin{x}}{x}\)
Condition 1: \(\frac{-1}{x} \leq \frac{\sin{x}}{x} \leq \frac{1}{x}\)
Condition 2: \(\lim_{x\rightarrow\infty} \frac{1}{x} = \lim_{x\rightarrow\infty} \frac{-1}{x} = \infty\)

Therfore \(\lim_{x\rightarrow\infty} \frac{\sin{x}}{x} = \lim_{x\rightarrow\infty} \frac{1}{x} = 0\)