<s>[INST] 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}
 [/INST] Hint: By multiplying the numerator and denominator by $(\sqrt{x+25}+5)(\sqrt{x+16}+4)$, and then using the difference of squares identity $(a-b)(a+b) = a^2-b^2$, we have:
$\dfrac{\sqrt{x+25}-5}{\sqrt{x+16}-4}$ $= \dfrac{(\sqrt{x+25}-5)(\sqrt{x+25}+5)(\sqrt{x+16}+4)}{(\sqrt{x+16}-4)(\sqrt{x+16}+4)(\sqrt{x+25}+5)}$ $= \dfrac{x(\sqrt{x+16}+4)}{x(\sqrt{x+25}+5)}$.
Can you take the limit of this expression as $x \to 0$?
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