91Ó°ÊÓ

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow 1}\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{6} $$

Explain why \(f\) is not continuous at \(a\). $$ f(x)=\left\\{\begin{array}{ll} \frac{1-\cos x}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0 \end{array} \quad a=0\right. $$

Find each limit, if it exists: |a) \(\lim _{x \rightarrow a^{-}} f(x)\) (b) \(\lim _{x \rightarrow a^{+}} f(x)\) (c) \(\lim _{x \rightarrow a} f(x)\) $$ f(x)=\frac{1}{x-8} ; \quad a=8 $$

Find the vertical and horizontal asymptotes for the graph of \(f\). $$ f(x)=\frac{3 x}{x^{2}+1} $$

Find all numbers at which \(f\) is discontinuous. $$ f(x)=\frac{3}{x^{2}+x-6} $$

Exercise \(27-32:\) Sketch the graph of tie piecewise-defined function \(f\) and, for the indicated value of \(a\), find each limit, if it exists: (a) \(\lim f(x)\) (b) \(\lim f(x)\) |c) \(\lim f(x)\) $$ f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x<1 \\ 2 & \text { if } x=1 \\ 4-x^{2} & \text { if } x>1 \end{array} \quad a=1\right. $$

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow 16} \frac{2 \sqrt{x}+x^{3 / 2}}{\sqrt[4]{x}+5} $$

Find the vertical and horizontal asymptotes for the graph of \(f\). $$ f(x)=\frac{1}{x^{3}+x^{2}-6 x} $$

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow-8} \frac{16 x^{2 / 3}}{4-x^{4 / 3}} $$

Exercise \(27-32:\) Sketch the graph of tie piecewise-defined function \(f\) and, for the indicated value of \(a\), find each limit, if it exists: (a) \(\lim f(x)\) (b) \(\lim f(x)\) |c) \(\lim f(x)\) $$ f(x)=\left\\{\begin{array}{ll} \frac{x^{4}+x}{x} & \text { if } x \neq 0 \\\ 2 & \text { if } x=0 \end{array} \quad a=0\right. $$