Answer
7
Work Step by Step
Theorem 3 - Squeeze Theorem
If $f(x) \leq g(x) \leq h(x)$ when $x$ is near $a$ (except possibly at a)
and $\displaystyle \lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}h(x)=L$, then$\quad \displaystyle \lim_{x\rightarrow a}g(x)=L$
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Our $f(x)$ is "squeezed" between$ u(x)=4x-9 $ and $v(x)=x^{2}-4x+7$
$u(x) \leq f(x) \leq v(x)$, for $x\geq 0$.
$\displaystyle \lim_{x\rightarrow 4}u(x)= \displaystyle \lim_{x\rightarrow 4}(4x-9)=4(4)-9=7$
$\displaystyle \lim_{x\rightarrow 4} v(x) \displaystyle \lim_{x\rightarrow 4}(x^{2}-4x+7)=4^{2}-4(4)+7=7$.
By the Squeeze Theorem,
$\displaystyle \lim_{x\rightarrow 4}f(x)=7$
