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
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 $$\lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}$$

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
$$\lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}$$
$$\Rightarrow\lim_{x\rightarrow2}\frac{(\sqrt{x+7}-3)(\sqrt{x+7}+3)}{(x-2)(\sqrt{x+7}+3)}$$
$$\Rightarrow\lim_{x\rightarrow2}\frac{(x+7-9)}{(x-2)(\sqrt{x+7}+3)}$$
$$\Rightarrow\lim_{x\rightarrow2}\frac{(x-2)}{(x-2)(\sqrt{x+7}+3)}$$
$$\Rightarrow\lim_{x\rightarrow2}\frac{1}{\sqrt{x+7}+3}$$
$$\Rightarrow\frac{1}{\sqrt{2+7}+3}\Rightarrow\frac{1}{\sqrt{9}+3}$$
$$\Rightarrow\frac{1}{3+3}$$
$$\Rightarrow\frac{1}{6}$$