91Ó°ÊÓ

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\left\\{\begin{array}{ll}{2 x+3,} & {x \leq 4} \\ {7+\frac{16}{x},} & {x>4}\end{array}\right. $$

Find the limits. $$ \lim _{x \rightarrow 0} e^{\sin x} $$

Find the limits. $$ \lim _{x \rightarrow-\infty} \frac{x+4 x^{3}}{1-x^{2}+7 x^{3}} $$

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) $$ \begin{array}{l}{\text { (i) the domain of } f \text { is }[-2,1]} \\ {\text { (ii) } f(-2)=f(0)=f(1)=0} \\ {\text { (iii) } \lim _{x \rightarrow-2^{+}} f(x)=2, \lim _{x \rightarrow 0} f(x)=0, \text { and }} \\ {\lim _{x \rightarrow 1^{-}} f(x)=1}\end{array} $$

Find the limits. $$ \lim _{y \rightarrow 6^{-}} \frac{y+6}{y^{2}-36} $$

Find the limits. $$ \lim _{x \rightarrow+\infty} \cos \left(2 \tan ^{-1} x\right) $$

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\left\\{\begin{array}{ll}{\frac{3}{x-1},} & {x \neq 1} \\ {3,} & {x=1}\end{array}\right. $$

Determine whether the statement is true or false. Explain your answer. If \(f(x)\) is continuous at \(x=c,\) then so is \(|f(x)|\)

Find the limits. $$ \lim _{y \rightarrow 6} \frac{y+6}{y^{2}-36} $$

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) $$ \begin{array}{l}{\text { (i) the domain of } f \text { is }(-\infty, 0]} \\\ {\text { (ii) } f(-2)=f(0)=1} \\ {\text { (iii) } \lim _{x \rightarrow-2} f(x)=+\infty}\end{array} $$