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}
$