91Ó°ÊÓ

Prove the $k=1$ case of the second part of Theorem $1.31$(b): that $\underset{x→∞}{\lim }{e}^{-x}=0$.

Use algebra, limit rules, and the continuity of $e^x$to prove that every exponential function of the form $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{kx}}$is continuous everywhere.

Prove that $\underset{\mathrm{θ}→0}{\lim }\frac{1-\cos \mathrm{θ}}{\mathrm{θ}}=0$ by using the double-angle identity $\cos 2\mathrm{θ}=1-2{\sin }^2\mathrm{θ}$ and the other special trigonometric limit $\underset{\mathrm{θ}→0}{\lim }\frac{\sin \mathrm{θ}}{\mathrm{θ}}=1$.

Use algebra, limit rules, and the continuity of $e^x$to prove that every exponential function of the form $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{A}{\mathit{b}}^{\mathbf{x}}$is continuous everywhere.

Use algebra, limit rules, and the continuity of $\ln x$on $(0,∞)$to prove that every logarithmic function of the form $f\left(x\right)={\log }_{b}x$ is continuous on$(0,∞)$.

Use algebra, limit rules, and the continuity of ln x on $(0,∞)$to prove that every logarithmic function of the form $f\left(x\right)={\log }_b\left(x\right)$is continuous on $(0,∞)$.

In the reading, we used the Squeeze Theorem to prove that $\underset{\mathbf{h}\mathbf{→}\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{h}\mathbf{=}\mathbf{0}$and $\underset{\mathbf{h}\mathbf{→}\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{h}\mathbf{=}\mathbf{1}$. Use these facts, the sum identity for cosine, and limit rules to prove thatrole="math" localid="1648152969592" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}$ is continuous everywhere.

Use the quotient rule for limits and the continuity of $\sin x$and $\cos x$to prove that $f\left(x\right)=\tan x$is continuous on its domain.

Use the quotient rule for limits and the continuity of $\cos x$to prove that $f\left(x\right)=secx$is continuous on its domain.

For each given function f, find a real number a that makes f continuous at $x=0,$if possible.

$$\begin{gathered} \left(a\right)f\left(x\right)=\left\{\begin{array}{ll}3x+1,& \mathrm{if}x<0\\ 2x+a,& \mathrm{if}xâ‰\yen 0\end{array}\right. \\ \left(b\right)f\left(x\right)=\left\{\begin{array}{ll}\frac{a}{x+2},& \mathrm{if}x<0\\ 3,& \mathrm{if}x=0\\ ax+1,& \mathrm{if}x>0\end{array}\right. \end{gathered}$$