# Find the limits: lim_{xrightarrow-2}frac{x+2}{sqrt{x^2+5}-3}

Find the limits:
$\underset{x\to -2}{lim}\frac{x+2}{\sqrt{{x}^{2}+5}-3}$
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Given limits:
$\underset{x\to -2}{lim}\frac{x+2}{\sqrt{{x}^{2}+5}-3}$
Rationalize the denominator, we get
$\underset{x\to -2}{lim}\frac{x+2}{\sqrt{{x}^{2}+5}-3}×\frac{\sqrt{{x}^{2}+5}+3}{\sqrt{{x}^{2}+5}+3}$
$\underset{x\to -2}{lim}\frac{\left(x+2\right)\left(\sqrt{{x}^{2}+5}+3\right)}{\left(\sqrt{{x}^{2}+5}{\right)}^{2}-{3}^{2}}$
$\underset{x\to -2}{lim}\frac{\left(x+2\right)\left(\sqrt{{x}^{2}+5}+3\right)}{{x}^{2}+5-9}$
$\underset{x\to -2}{lim}\frac{\left(x+2\right)\left(\sqrt{{x}^{2}+5}+3\right)}{{x}^{2}-4}$
$\underset{x\to -2}{lim}\frac{\left(x+2\right)\left(\sqrt{{x}^{2}+5}+3\right)}{{x}^{2}-{2}^{2}}$
$\underset{x\to -2}{lim}\frac{\left(x+2\right)\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)\left(x+2\right)}$
$\underset{x\to -2}{lim}\frac{\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)}$
Apply the limits $x=-2$, we get
$\underset{x\to -2}{lim}\frac{\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)}=\frac{\left(\sqrt{\left(-2{\right)}^{2}+5}+3}{\left(-2-2\right)}$
$\underset{x\to -2}{lim}\frac{\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)}=\frac{\left(\sqrt{4+5}+3}{\left(-2-2\right)}$
$\underset{x\to -2}{lim}\frac{\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)}=\frac{\left(3+3\right)}{\left(-2-2\right)}$
$\underset{x\to -2}{lim}\frac{\left(\sqrt{{x}^{2}+5}+3\right)}{\left(x-2\right)}=-\frac{3}{2}$