Question

a) Using limit rules each of the following limits \lim_{x\rightarrow0}\frac{x+1}{x} \lim_{x\rightarrow0}\frac{x+1}{x^{2}} \lim_{x\rightarrow0}\frac{x}{x+1} \lim_{x\rightarrow0}\frac{x+1}{x+2} b) Using limit rules evaluate \lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}

Limits and continuity
ANSWERED
asked 2021-05-27
a) Using limit rules each of the following limits
\(\lim_{x\rightarrow0}\frac{x+1}{x}\)
\(\lim_{x\rightarrow0}\frac{x+1}{x^{2}}\)
\(\lim_{x\rightarrow0}\frac{x}{x+1}\)
\(\lim_{x\rightarrow0}\frac{x+1}{x+2}\)
b) Using limit rules evaluate \(\lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}\)

Expert Answers (1)

2021-05-28
Step 1
\(\lim_{x\rightarrow0}\frac{x+1}{x}\Rightarrow\frac{0+1}{0}\Rightarrow\frac{1}{0}\Rightarrow\infty\)
\(\lim_{x\rightarrow0}\frac{x+1}{x^{2}}\Rightarrow\frac{0+1}{0^{2}}\Rightarrow\frac{1}{0}\Rightarrow\infty\)
\(\lim_{x\rightarrow0}\frac{x}{x+1}\Rightarrow\frac{0}{0+1}\Rightarrow\frac{0}{1}\Rightarrow0\)
\(\lim_{x\rightarrow0}\frac{x+1}{x+2}\Rightarrow\frac{0+1}{0+2}\Rightarrow\frac{1}{2}\)
Step 2
\(\lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}\)
\(\Rightarrow\lim_{x\rightarrow2}\frac{(\sqrt{x+7}-3)(\sqrt{x+7}+3)}{(x-2)(\sqrt{x+7}+3)}\)
\(\Rightarrow\lim_{x\rightarrow2}\frac{(x+7-9)}{(x-2)(\sqrt{x+7}+3)}\)
\(\Rightarrow\lim_{x\rightarrow2}\frac{(x-2)}{(x-2)(\sqrt{x+7}+3)}\)
\(\Rightarrow\lim_{x\rightarrow2}\frac{1}{\sqrt{x+7}+3}\)
\(\Rightarrow\frac{1}{\sqrt{2+7}+3}\Rightarrow\frac{1}{\sqrt{9}+3}\)
\(\Rightarrow\frac{1}{3+3}\)
\(\Rightarrow\frac{1}{6}\)
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