<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] If you want to go even beyond the limit itself, let $x=y+4$ to make
$$A=\frac{\sqrt{2 x+1}-3}{\sqrt{x}-2}=\frac{\sqrt{2 y+9}-3}{\sqrt{y+4}-2}$$ and use the binomial expansion or (better) Taylor series around $y=0$. This would give
$$A=\frac{\frac{y}{3}-\frac{y^2}{54}+O\left(y^3\right) }{\frac{y}{4}-\frac{y^2}{64}+O\left(y^3\right)}=\frac{4}{3}+\frac{y}{108}+O\left(y^2\right)$$ Back to $x$,
$$A=\frac{4}{3}+\frac{x-4}{108}+O\left((x-4)^2\right)$$
Try using $x=5$; the exact value is $A=\left(2+\sqrt{5}\right) \left(\sqrt{11}-3\right)\approx 1.34124$ while the expansion gives $\frac{145}{108}\approx 1.34259$.
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