The limit of $(4-\sqrt{16-7\sin(x)})/(8x)$ at zero without using L'Hôpital I stumbled across this silly limit and I am perplexed at how I can arrive to a solution by only relying on the simplest rules of limits.
$$
\lim_{x \to 0}\frac{4-\sqrt{16-7\sin(x)}}{8x}
$$
Any help is appreciated, thanks in advance.

$$
\begin{aligned}
\lim _{x\to 0}\left(\frac{4-\sqrt{16-7\sin \left(x\right)}}{8x}\right)
& =\lim _{x\to 0}\left(\frac{4-\left(4-\frac{7x}{8}+o\left(x\right)\right)}{8x}\right)
\\& = \color{red}{\frac{7}{64}}
\end{aligned}
$$
Solved with Taylor expansion
