# The Squeeze Theorem $ \begin{align} \text{If} \, &g(x) \le f(x) \le h(x) \, \forall x \in D \\ \text{and} \, &\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\\ \text{then} \, &\lim_{x \to a} f(x) = L \end{align} $ Interesting thing to note here is that the domain of the three functions can be freely defined. This allows us to limit the domain to be 'around $a, in order to satisfy the first requirement. The squeeze theorem is used to calculate [[Limit|limits]] that are difficult to calculate using other methods. It is used to [[Derivatives of Sine and Cosine|determine derivatives of trig functions]]. ## Using The Squeeze Theorem To Evaluate Limits of Trigonometric Functions Limits of complex trigonometric expressions can be determined by building them up from the fact that the range of trig functions is limited. Take $5 + 2x^2 \sin \left( \frac 1 {x^7} \right)$ as an example (ignoring for the moment the fact that its limit as $x \to 0$ can be easily determined by splitting up the individual parts): $ \begin{gather} -1 \le \sin \frac 1 {x^7} \le 1 & \text{multiply by $2x^2$} \\ -2x^2 \le 2x^2 \sin \frac 1 {x^7} \le 2x^2 & \text{add 5} \\ 5-2x^2 \le 5+ 2x^2 \sin \frac 1 {x^7} \le 5+ 2x^2 \\ \lim_{x\to 0} \left[5 - 2x^2\right] = \lim_{x\to0} \left[5+2x^2\right] = 5 \Downarrow \\ \lim_{x\to0} \left[ 5 +2x^2\sin \frac 1 {x^7} \right] = 5 \end{gather} $ ## Sources - [Brilliant](https://brilliant.org/wiki/squeeze-theorem/)