True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the limit of $f(x)$ as $x$ approaches $c$ is $0,$ then there must exist a number $k$ such that $f(k)<0.001$

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## Recommended Questions

True or False? In Exercises $73-76,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f$ is undefined at $x=c$ , then the limit of $f(x)$ as $x$ approaches $c$ does not exist.

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$\lim _{x \rightarrow c} f(x)=L,$ then $f(c)=L$$

TRUE OR FALSE? In Exercises 59-62, determine whether the statement is true or false. Justify your answer.

If $f(x)$ increases without bound as $x$ approaches $c$, then the limit of $f(x)$ exists.

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$f(c)=L,$ then $\lim _{\mathrm{lim}} f(x)=L$$

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$\lim _{x \rightarrow \infty} f(x)=L, \text { then } \int_{0}^{\infty} f(x) d x$$

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

The limit of a function as $x$ approaches $c$ does not exist if the function approaches $-3$ from the left of $c$ and $3$ from the right of $c$.

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$\int f(x) g(x) d x=\left(\int f(x) d x\right)\left(\int g(x) d x\right)$

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $F(x)$ and $G(x)$ are antiderivatives of $f(x),$ then

$$F(x)=G(x)+C$$

True or False? In Exercises $83-86$ , determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

If $f^{\prime \prime}(x)<0$ for all real numbers $x,$ then $f$ decreases without

bound.

TRUE OR FALSE? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer.

If $f(c) = L$, then $\lim_{x \to c}\ f(x) = L$

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { If } f \text { is continuous on }[0, \infty) \text { and } \lim _{x \rightarrow \infty} f(x)=0, \text { then } \int_{0}^{\infty} f(x) d x} \\ {\text { converges. }}\end{array}$$

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f(x)=-g(x)+b,$ then $f^{\prime}(x)=-g^{\prime}(x)$

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { If } f \text { is continuous on }[0, \infty) \text { and } \int_{0}^{\infty} f(x) d x \text { diverges, then }} \\ {\lim _{x \rightarrow \infty} f(x) \neq 0}\end{array}$$

True or False? In Exercises $77-79$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y$ is a function of $t$ and $x$ is a function of $t,$ then $y$ is a function of $x .$

True or False? In Exercises $95-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false

$\lim _{x \rightarrow 0} \frac{x^{2}+x+1}{x}=\lim _{x \rightarrow 0} \frac{2 x+1}{1}=1$

True or False? In Exercises $63-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\int_{a}^{b} f(x) g(x) d x=\left[\int_{a}^{b} f(x) d x\right]\left[\int_{a}^{b} g(x) d x\right]$$

True or False? In Exercises $83-86$ , determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.

The slope of the graph of the inverse tangent function is positive for all $x.$

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c>0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

True or False? In Exercises $95-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false

An indeterminate form does not guarantee the existence of a limit.

TRUE OR FALSE? In Exercises 59-62, determine whether the statement is true or false. Justify your answer.

When the degrees of the numerator and denominator of a rational function are equal, the limit does not exist.

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $p(x)$ is a polynomial function, then $p$ has exactly one antiderivative whose graph contains the origin.

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { If } f^{\prime} \text { is continuous on }[0, \infty) \text { and } \lim _{x \rightarrow \infty} f(x)=0, \text { then }} \\ {\int_{0}^{\infty} f^{\prime}(x) d x=-f(0)}\end{array}$$

True or False? In Exercises 65-68, determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

The graph of a function cannot cross a vertical asymptote.

True or False? In Exercises $83 - 86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of $f$ and $g$ is $1 ,$ then the area of the region bounded by the graphs of

$h ( x ) = f ( x ) + C$ and $k ( x ) = g ( x ) + C$ is also $1 .$

True or False? In Exercises $95-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

A limit of the form $\infty / 0$ is indeterminate.

True or False? In Exercises $95-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false

If $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=1,$ then $\lim _{x \rightarrow \infty}[f(x)-g(x)]=0$

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Each antiderivative of an $n$ th-degree polynomial function is an $(n+1)$ th-degree polynomial function.

True or False? In Exercises $83-86$ , determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.

The range of $$y=\arcsin x [0, \pi]$$

TRUE OR FALSE? In Exercises 59-62, determine whether the statement is true or false. Justify your answer.

If a sequence converges, then it has a limit.

True or False? In Exercises 73 and $74,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f$ is continuous and nonnegative on $[a, b],$ then the limits as $n \rightarrow \infty$ of its lower sum $s(n)$ and upper sum $S(n)$ both exist and are equal.

True or False? In Exercises $49-53,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=f(x), f$ is increasing and differentiable, and $\Delta x>0,$

then $\Delta y \geq d y .$

True or False? In Exercises $105-110$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$\lim _{x \rightarrow c} f(x)=L$$ and $f(c)=L,$ then $f$ is continuous at $c$

True or False? In Exercises $91-96,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $\lim _{n \rightarrow \infty} a_{n}=0,$ then $\sum_{n=1}^{\infty} a_{n}$ converges.

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Amplitude is always positive

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

The limit of the product of two functions is equal to the product of the limits of the two functions.

TRUE OR FALSE? In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer.

If $f(x) = x + 1$ and $g(x) = 6x$, then $(f \circ g)(x) = (g \circ f)(x)$.

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f(x)=0,$ then $f^{\prime}(x)$ is undefined.

True or False? In Exercises $63-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\int_{a}^{b}[f(x)+g(x)] d x=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x$$

True or False? In Exercises $83 - 86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If

$$\int _ { a } ^ { b } [ f ( x ) - g ( x ) ] d x = A$$

then

$$\int _ { a } ^ { b } [ g ( x ) - f ( x ) ] d x = - A$$

True or False? In Exercises $83-86$ , determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

Every polynomial function has an absolute maximum and an

absolute minimum on $(-\infty, \infty).$

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.$\begin{array}{l}{\text { The graph of every cubic polynomial has precisely one point }} \\ {\text { of inflection. }}\end{array}$

TRUE OR FALSE? In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer.

The slope of the graph of $y=x^2$ is different at every point on the graph of $f$.

True or False? In Exercises $91-96$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

There is no function with an infinite number of critical points.

TRUE OR FALSE? In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer.

If you are given two functions $f(x)$ and $g(x)$, you can calculate $(f \circ g)(x)$ if and only if the range of $g$ is a subset of the domain of $f$.

True or False? In Exercises $115-120$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false..

$$\lim _{x \rightarrow 2} f(x)=3,$$ where $$f(x)=\left\{\begin{array}{ll}{3,} & {x \leq 2} \\ {0,} & {x>2}\end{array}\right.$$

True or False? In Exercises $83-86$ , determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

Every rational function has a slant asymptote.

True or False? In Exercises $95-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false

A limit of the form $\infty . \infty$ is indeterminate.

True or False? In Exercises $91-96$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The function $f(x)=x$ has no extrema on any open interval.

True or False? In Exercises $115-120$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false..

If $$\lim _{x \rightarrow c} f(x)=L,$$ then $f(c)=L$

TRUE OR FALSE? In Exercises 73-76, determine whether the statement is true or false. Justify your answer.

If $D\neq0$ and $E\neq0$, then the graph of $x^2-y^2+Dx+Ey=0$ is a hyperbola.

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

If $y$ is a function of $t$, and $x$ is a function of $t$, then $y$ must be a function of $x$.

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\int_{0}^{\infty} e^{a x} d x \text { converges for } a<0$$

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c=0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

has only one $x$ -intercept.

True or False? In Exercises $49-53,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y$ is differentiable, then$$\lim _{\Delta x \rightarrow 0}(\Delta y-d y)=0$$

True or False $\lim f(x)$ exists and equals some number for any function $f$ as long as $c$ is in the domain of $f$

True or False? In Exercises $115-120$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false..

$$\lim _{x \rightarrow 0} \frac{|x|}{x}=1$$

$$\lim _{x \rightarrow 0} \frac{|x|}{x}=1$$

True or False? In Exercises $115-120$ , determine whether

the statement is true or false. If it is false, explain why or give

an example that shows it is false.

If $f(x) < g(x)$ for all $x \neq a,$ then $$\lim _{x \rightarrow \infty} f(x) < \lim _{x \rightarrow n} g(x)$$

TRUE OR FALSE? In Exercises 73-76, determine whether the statement is true or false. Justify your answer.

If the asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, where $a, b > 0$, intersect at right angles, then $a=b$.

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum of $y=x^{2}$ on the open interval $(-3,3)$ is 9

True or False? In Exercises $83-86$ , determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.

$$\frac{d}{d x}[\arctan (\tan x)]=1$$ for all $x$ in the domain.

True or False? In Exercises 83-86, determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

$$\begin{array}{l}{\text { If } f^{\prime \prime}(x)>0 \text { for all real numbers } x, \text { then } f \text { increases without }} \\ {\text { bound. }}\end{array}$$

TRUE OR FALSE? In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer.

A tangent line to a graph can intersect the graph only at the point of tangency.

True or False? In Exercises $49-53,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The tangent line approximation at any point for any linear equation is the linear equation itself.

TRUE OR FALSE? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer.

If $f$ is an even function, then $f^{-1}$ exists.