Question

Solve

${lim}_{x\to 0}\frac{\sqrt{x+4}-2}{\sqrt{x+9}-3}$

Answer 1

${lim}_{x\to 0}\frac{\sqrt{x+4}-2}{\sqrt{x+9}-3}$

rationalize the denominator

$={lim}_{x\to 0}\left(\frac{(\sqrt{x+4}-2)(\sqrt{x+9}+3)}{x}\right)$

multiply by the conjugate of $\sqrt{x+4}-2:\frac{x}{\sqrt{x+4}+2}$

$={lim}_{x\to 0}\left(\frac{\frac{x}{\sqrt{x+4}+2}(\sqrt{x+9}+3)}{x}\right)$

$={lim}_{x\to 0}\left(\frac{\sqrt{x+9}+3}{\sqrt{x+4}+2}\right)$

$=\frac{\sqrt{0+9}+3}{\sqrt{0+4}+2}$

$=\frac{3}{2}$