Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 36

Suppose $2x \leq g(x) \leq x^4 - x^2 + 2$ for all $x$, evaluate $\lim \limits_{x \to 1} g(x)$

$\begin{equation} \begin{aligned} & \text{Using Squeeze Theorem}\\ \phantom{x}& && \lim \limits_{x \to 1} 2x \leq \lim \limits_{x \to 1} g(x) \leq \lim \limits_{x \to 1} (x^4-x^2+2)\\ \phantom{x}& && 2(1) \leq \lim \limits_{x \to 1} g(x) \leq [(1)^4 - (1)^2 +2]\\ \phantom{x}& && 2 \leq \lim \limits_{x \to 1} g(x) \leq 2\\ & \text{We have,}\\ \phantom{x}& && \lim \limits_{x \to 1} 2x = \lim \limits_{x \to 1} (x^4-x^2+2)\\ & \text{Therefore,}\\ \phantom{x}& && \lim \limits_{x \to 1} g(x) = 2 \end{aligned} \end{equation}$