Question

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

Limits and continuity
ANSWERED
asked 2021-03-09
Find the limits:
\(\lim_{x\rightarrow-2}\frac{x+2}{\sqrt{x^2+5}-3}\)

Answers (1)

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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