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}

Note that, 
$$
\lim_{x \to 0}\dfrac{\sqrt{x + a^2} - a}{x} = \lim_{x \to 0}\dfrac{1}{\sqrt{x + a^2} + a} = \dfrac{1}{2a} 
$$
for $a > 0$. Thus,
\begin{eqnarray}
\\&\lim_{x\to 0}\frac{\sqrt{x+25}-5} {\sqrt{x+16}-4} = \dfrac{\lim_{x \to 0}\dfrac{\sqrt{x + 25} - 5}{x}}{\lim_{x \to 0}\dfrac{\sqrt{x + 16} - 4}{x}} = \dfrac{1/10}{1/8} = \dfrac{4}{5}
\end{eqnarray}
