<s>[INST] Find the limit of $\begin{equation*} \lim_{x \rightarrow 4} \frac{\sqrt{1 + 2x} -3}{\sqrt{x} - 2} \end{equation*}$ 
Find the following limit: \begin{equation*} \lim_{x \rightarrow 4} \frac{\sqrt{1 + 2x} -3}{\sqrt{x} - 2} \end{equation*}

I have tried to divide the numerator and denominator by $\sqrt{x}$, but it did not work.
I have tried to multiply by the conjugates of the numerator and denominator simultaneously but it did not work. 
I have tried to multiply by the conjugates of the numerator only but it did not work.
So what shall I do?
 [/INST] Using a substitution and a derivative:
Set $x=t^2$:
$$ \lim_{x \rightarrow 4} \frac{\sqrt{1 + 2x} -3}{\sqrt{x} - 2} = \lim_{t \rightarrow 2} \frac{\sqrt{1 + 2t^2} -3}{t - 2} = f'(2) \mbox{ for } f(t) = \sqrt{1 + 2t^2}$$
So, 
$$f'(t) = \frac{2t}{\sqrt{1 + 2t^2}} \Rightarrow \lim_{x \rightarrow 4} \frac{\sqrt{1 + 2x} -3}{\sqrt{x} - 2} = f'(2) = \frac{4}{3}$$
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