<s>[INST] Help calculating a math limit Can anyone help with this limit?
\begin{equation*}
\lim_{x \rightarrow 4}
\frac{16\sqrt{x-\sqrt{x}}-3\sqrt{2}x-4\sqrt{2}}{16(x-4)^2}
\end{equation*}
I've tried a variable change of  \begin{equation*} y=\sqrt{x} \end{equation*} but this didn't help.
 [/INST] The variable change is a good idea to begin with, but I'd use $2y=\sqrt{x}$
$$
\lim_{y\to 1}
\frac{16\sqrt{4y^2-2y}-12\sqrt{2}\,y^2-4\sqrt{2}}{16(4y^2-4)^2}=
\frac{4\sqrt{2}}{256}
\lim_{y\to 1}
\frac{4\sqrt{2y^2-y}-(3y^2+1)}{(y^2-1)^2}
$$
You can do this with a Taylor expansion or by “rationalizing”: note that
$$
16(2y^2-1)-(3y^2+1)^2=-9y^4+26y^2-17=-(y^2-1)(9y^2-17)
$$
</s>