<s>[INST] If $f$ is differentiable at $4$ and $f^\prime(4) = 5$, evaluate the following limit: $\lim_{x \to 4} \frac{f(x)-f(4)}{\sqrt{x} - 2}$. If $f$ is differentiable at $4$ and $f^\prime(4) = 5$, evaluate the following limit: $\lim_{x \to 4} \frac{f(x)-f(4)}{\sqrt{x} - 2}$.
\begin{align*}
\\ h &= \sqrt{x} - 2
\\
\\ h \to 0 &\text{ as } x \to 4
\\
\\ \lim_{x \to 4} \frac{f(x)-f(4)}{\sqrt{x} - 2} &= \lim_{h \to 0} \frac{f(x)-f(4)}{h}
\\ \lim_{h \to 0} \frac{f(4-h)-f(x)}{-h} &\implies \lim_{h \to 0} \frac{f(4-h)-f(4)}{-h} = f^\prime(4) = 5
\end{align*}
 [/INST] $$\lim_{x \to 4} \frac{f(x)-f(4)}{\sqrt{x} - 2}$$
$$=\lim_{x \to 4} \frac{f(x)-f(4)}{x-4}(\sqrt x+2)$$
$$=f'(4)\cdot\lim_{x \to 4}(\sqrt x+2)=\cdots$$
</s>