<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] you can do this. $$f(4 + h) = f(4) + 5h+\cdots, \sqrt{4+h} = (4+h)^{1/2} = 4^{1/2} + \frac12 4^{-1/2}h+\cdots \\\to f(4+h) - f(4) = 5h+\cdots, \sqrt{4+h} = 2 +\frac 14 h+\cdots  \\
\to \frac{f(4+h) - f(4)}{\sqrt{4+h} - 2 }= 20+\cdots$$ so the limit is $20.$
</s>