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

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

Question
Limits and continuity
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
\(\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(\sqrt{(-2)^2+5}+3}{(-2-2)}\)
\(\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(\sqrt{4+5}+3}{(-2-2)}\)
\(\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=\frac{(3+3)}{(-2-2)}\)
\(\lim_{x\rightarrow-2}\frac{(\sqrt{x^2+5}+3)}{(x-2)}=-\frac{3}{2}\)
0

Relevant Questions

asked 2020-10-23
Find the limits
\(\lim_{x\rightarrow-3}\frac{2-\sqrt{x^2-5}}{x+3}\)
asked 2020-12-21
Find the limits.
\(\lim_{x\rightarrow-\infty}\frac{2^x+4^x}{5^x-2^x}\)
asked 2021-02-18
Evaluate the following limits.
\(\lim_{x\rightarrow-3}\frac{\sin(x+3)}{x^2+8x+15}\)
asked 2021-02-25
Find the following limits or state that they do not exist. \(\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}\)
asked 2021-02-12
Find the following limit.
\(\lim_{x\rightarrow+\infty}(\frac{3-12x}{3x+5}+\csc\frac{\pi x+\pi}{4x})\)
asked 2021-03-11
Evaluate the following limits.
\(\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}\)
asked 2020-11-14
Find the limits. \(\lim_{x\rightarrow-2} g(f(x))\)
\(f(x)=x+7 g(x)=x^2\)
asked 2021-02-25
Find the limit
\(\lim_{x\rightarrow-\infty}\frac{x^2-4x+8}{3x^3}\)
asked 2020-12-16
Find the limits: \(\lim_{x\rightarrow4}\frac{4x-x^2}{2-\sqrt{x}}\)
asked 2020-11-10
Find the limits:
\(\lim_{x\rightarrow2}\frac{\sqrt{x^2+12}-4}{x-2}\)
...