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

Question
Limits and continuity
Find the limits:
$$\lim_{x\rightarrow-2}\frac{x+2}{\sqrt{x^2+5}-3}$$

2021-03-10
Given limits:
$$\lim_{x\rightarrow-2}\frac{x+2}{\sqrt{x^2+5}-3}$$
Rationalize the denominator, we get
$$\lim_{x\rightarrow-2}\frac{x+2}{\sqrt{x^2+5}-3}\times\frac{\sqrt{x^2+5}+3}{\sqrt{x^2+5}+3}$$
$$\lim_{x\rightarrow-2}\frac{(x+2)(\sqrt{x^2+5}+3)}{(\sqrt{x^2+5})^2-3^2}$$
$$\lim_{x\rightarrow-2}\frac{(x+2)(\sqrt{x^2+5}+3)}{x^2+5-9}$$
$$\lim_{x\rightarrow-2}\frac{(x+2)(\sqrt{x^2+5}+3)}{x^2-4}$$
$$\lim_{x\rightarrow-2}\frac{(x+2)(\sqrt{x^2+5}+3)}{x^2-2^2}$$
$$\lim_{x\rightarrow-2}\frac{(x+2)(\sqrt{x^2+5}+3)}{(x-2)(x+2)}$$
$$\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}$$
Apply the limits x=-2, we get
$$\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(\sqrt{(-2)^2+5}+3}{(-2-2)}$$
$$\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(\sqrt{4+5}+3}{(-2-2)}$$
$$\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(3+3)}{(-2-2)}$$
$$\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=-\frac{3}{2}$$

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