How to solve $\displaystyle\lim_{x\to 0}\tfrac{\sqrt{x+25}-5} {\sqrt{x+16}-4}$ \begin{eqnarray}
\\&\lim_{x\to 0}\frac{\sqrt{x+25}-5} {\sqrt{x+16}-4}
\end{eqnarray}
Undefined limit
\begin{eqnarray}
\frac{0} {0}
\end{eqnarray}

METHOD 1:
One quick way is to write
$$\sqrt{x+a^2}=|a|\sqrt{1+x/a^2}=|a|\,\left(1+\frac12 \left(\frac{x}{a^2}\right)+O\left(x\right)^2\right)$$
Then, we have
$$\frac{\sqrt{x+25}-5}{\sqrt{x+16}-4}=\frac45 +O(x)\to \frac45$$

METHOD 2:
Using L'Hospital's Rule gives
$$\lim_{x\to 0}\frac{\sqrt{x+25}-5}{\sqrt{x+16}-4}=\lim_{x\to 0}\frac{\frac{1}{2\sqrt{x+25}}}{\frac{1}{2\sqrt{x+16}}}=\frac45$$
