Limits

$\underset{x\to 0}{\lim }\frac{\sin x}{\cos x}$

Calculate each of the limits:

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{cot x}{\cos x}$.

Calculate each of the limits:

$\underset{x\to 1}{\lim }x{\sin }^{-1}\frac{x}{2}$.

Calculate each of the limits:

$\underset{x\to 3}{\lim }\frac{\sqrt{x}}{{\tan }^{-1}\sqrt{x}}$

Calculate each of limits:

$\underset{x\to 1}{\lim }\frac{1}{{\sin }^{-1}x}$.

Calculate each of the limits:

$\underset{x\to 0}{\lim }\frac{1}{se{c}^{-1}x}$.

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(3+h)}^2-3^2}{h}$.

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(2+h)}^2-2^2}{h}$.

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(1+h)}^3-1^3}{h}$.

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(-1+h)}^3-{(-1)}^3}{h}$.

Describe the intervals on which each function f is continuous. At each point where f fails to be

continuous, use limits to determine the type of discontinuity

and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-3x-1, if x\ne -2\\ -3, if x=-2\end{array}\right.$

Describe the intervals on which each function f  is continuous. At each point where f fails to be

continuous, use limits to determine the type of discontinuity

and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{l}\frac{3x^2}{4+x}, if x<2\\ x^2-2, if x\ge 2\end{array}\right.$

Describe the intervals on which each function f  is continuous. At each point where f fails to be

continuous, use limits to determine the type of discontinuity

and any left- or right-continuity.

Describe the intervals on which each function f  is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{ll}x+1,& if x<3\\ 2,& if x=3\\ x^2-9,& if x>3\end{array}\right.$

Describe the intervals on which each function f  is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{ll}\sin x,& if x<\mathrm{\pi }\\ \cos x,& if x\ge \mathrm{\pi }\end{array}\right.$

Describe the intervals on which each function f  is continuous. At each point where f  fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{ll}{2}^x-1,& if x\le 0\\ \frac{4^x-1}{2^x-1}& if x>0\end{array}\right.$

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{x\to 0}{\lim }x \sin \frac{1}{x}$

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{x\to 0}{\lim }x \sin \frac{1}{x^2}$

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{x\to 0}{\lim }\left(e^x-1\right) \sin \frac{1}{x}$

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{x\to 0}{\lim } \sin x \sin \frac{1}{x}$

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}$

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}$

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{ }\mathbf{lim}\mathbf{ }}\mathit{x}{\mathbf{ }\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\frac{\mathbf{1}}{\mathbf{x}}$

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{ }\mathbf{lim}\mathbf{ }}{\mathit{x}}^{\mathbf{2}}{\mathbf{ }\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\frac{\mathbf{1}}{\mathbf{x}}$

You constructed a piecewise-defined function from the $2000$ Federal Tax Rate Schedule that you will use in the next two problems. Specifically, you found that a person who makes m dollars a year will pay $T\left(m\right)$ dollars in tax, given by the function

$\begin{array}{r}0.15m,\text{ if }0\le m\le 26,250\\ 3,937+0.28(m-26,250),\text{ if }26,250<m\le 63,550\\ 14,381+0.31(m-63,550),\text{ if }63,550<m\le 132,600\\ 35,787+0.36(m-132,600),\text{ if }132,600<m\le 288,350\\ 91,857+0.396(m-288,350),\text{ if }m>288,350\end{array}$

Suppose you make $\$63,550$ a year and pay tax according to the given formula.

(a)  Calculate the value of T(63,550) and the limit of $T\left(m\right)$as m approaches 63,550 from the left and from the right.

(b)  Use part $\left(a\right)$ to argue that the function $T\left(m\right)$ is continuous at $m=63,550$. What does this mean in real-world terms?

you constructed a piecewise-defined function from the $2000$ Federal Tax Rate Schedule that you will use in the next two problems. Specifically, you found that a person who makes m dollars a year will pay $T\left(m\right)$ dollars in tax, given by the function.

$\begin{array}{r}0.15m,\text{ if }0\le m\le 26,250\\ 3,937+0.28(m-26,250),\text{ if }26,250<m\le 63,550\\ 14,381+0.31(m-63,550),\text{ if }63,550<m\le 132,600\\ 35,787+0.36(m-132,600),\text{ if }132,600<m\le 288,350\\ 91,857+0.396(m-288,350),\text{ if }m>288,350\end{array}$

Suppose you make $\$288,350$ a year and pay taxes according to the given formula.

(a)  Calculate the value of T(288,350) and the limit of $T\left(m\right)$ as $m$ approaches $288,350$ from the left and from the right.

(b)  Use part $\left(a\right)$ to argue that the function T(m) is continuous at $m=288,350$. What does this mean in real-world terms?

Calculate the derivative of $f\left(x\right)=x^2$ at $c=0$.

Calculate the derivative of $f\left(x\right)=x^2$ at  $c=2$.

Calculate the derivative of $f\left(x\right)=x^2$ at  $c=4$.

Sketch a graph of $f\left(x\right)=x^2$, and sketch the lines that point in the direction of the curve at $(0, f(0\left)\right), (2, f(2\left)\right)$, and $(4, f(4\left)\right)$. Relate the slopes of these lines to the answers to the last three exercises.

Use the definition of the derivative to calculate the derivative of $f\left(x\right)=x^{-1/2}$ at $c = 4$. As in the previous calculation, you will need to multiply numerator and denominator by a conjugate at some point.

Calculate the derivative of $f\left(x\right)=e^x$ at $c = 0$. At some point you should need the characterization of e given in Theorem 1.26.

Use limit rules and the continuity of polynomial functions to prove that every rational function is continuous on its domain.

Prove the constant multiple rule for limits:
$\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{L}\mathbf{\text{ and }}\mathit{k}\mathbf{\in }\mathbf{ℝ}\mathbf{,}\mathbf{\text{ then }}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{k}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{L}$

Prove the difference rule for limits by applying the sum and constant multiple rules for limits.

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.

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 $(0,\infty )$ to prove that every logarithmic function of the form $f\left(x\right)={\log }_b\left(x\right)$ is continuous on $(0,\infty )$.

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

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

Use the composition rule for limits and the fact that $\tan x$ is continuous on its domain to prove that ${\tan }^{-1}x$ is continuous everywhere.

Read the section and make your own sum-

mary of the material.

Determine whether each function approaches 0, approaches a nonzero real number, or becomes infinite as x approaches each indicated value.

$$\begin{gathered} f\left(x\right)=csc x, \mathrm{with} x\to 0 \mathrm{and} x\to \pi . \\ f\left(x\right)={\tan }^2 x, \mathrm{with} x\to 0 \mathrm{and} x\to \pi . \\ f\left(x\right)={\sin }^{-1} x, \mathrm{with} x\to 0 \mathrm{and} x\to 1. \\ f\left(x\right)={\tan }^{-1}\sqrt{x}, \mathrm{with} x\to 0 \mathrm{and} x\to 3. \end{gathered}$$

Construct Examples

The definition of infinite limits and limits at infinity: Write each limit statement that follows in terms of the formal definition of limit. Then approximate the largest value of $\delta or N$corresponding to $\in =0.5 or M=100$, as appropriate, and illustrate this choice of $\delta or N$  on a graph of $f$.

   $\underset{x\to \mathrm{\infty }}{lim} \frac{2x}{x-1}=2$

In Exercises 3–6, $\underset{x\to c}{\lim }f\left(x\right)=L$ and $\underset{x\to c}{\lim }g\left(x\right)=M$ for some real numbers L and M. What, if anything, can you say about $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ in each case?

$L\ne 0$ and $M\ne 0$.

In Exercises 3–6, $\underset{x\to c}{\lim }f\left(x\right)=L$and $\underset{x\to c}{\lim }g\left(x\right)=M$for some real numbers L and M. What, if anything, can you say about  $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ in each case?

$L=0$ and $M\ne 0$