Question 1

Is $f ( x ) = \sqrt { x }$ continuous at $x = - 1 ?$ Explain why or why not.

Accepted Answer

It is not continuous at x = -1 because it is not defined at x = -1.

Question 2

$f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } + 1 & \text { if } x < - 1 \
3 & \text { if } - 1 \leq x < 2 \
2 x - 1 & \text { if } x \geq 2
\end{array} \right.$ Find $\lim _ { x \rightarrow 2 } f ( x )$ if it exists.

Accepted Answer

A)  Does not exist 
B)  2 
C)  0 
D)  3 
A)  Does not exist 
B)  2 
C)  0 
D)  3 D

Question 3

$f ( x ) = \left\{ \begin{array} { c } 
x ^ { 2 } \text { if } x < 3 \
2 x + 2 \text { if } x \geq 3
\end{array} \right.$ Find $\lim _ { x \rightarrow 3 } f ( x )$ if it exists. Otherwise, explain by one-sided limits.

Accepted Answer

$\lim _ { x \rightarrow 3 } f ( x )$ does not exist because $\lim f ( x ) = 9 , \text { but } \lim f ( x ) = 8$

Question 4

Use the Intermediate Value Theorem to find a $c \in ( - \infty , \infty )$ for $f ( x ) = x ^ { 3 } + 2$ on the interval [-2, 3] such that $f ( c ) = 1$

Accepted Answer

The answer of Use the Intermediate Value Theorem to find...

Question 5

Find $\lim _ { x \rightarrow + \infty } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }$

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow +...

Question 6

Find $\lim _ { x \rightarrow - \infty } \frac { 1 - 2 x } { \sqrt { 4 x ^ { 2 } - 5 } }$

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow -...

Question 7

Given $f ( x ) = \frac { 4 x } { x ^ { 2 } + 3 x }$ , find all the $X$ values where $f$ is not continuous.

Accepted Answer

A)  - 3 
B)  0 
C)  0 and - 3 
D)  - 3 and 3 
A)  - 3 
B)  0 
C)  0 and - 3 
D)  - 3 and 3

Question 8

Find the equation of a possible function $f$ with $\lim f ( x ) = 0$ and $\lim _ { x \rightarrow + \infty } f ( x ) = 0$ not defined at $x = 3$ , and $f ( 0 ) = - 1$

Accepted Answer

The answer of Find the equation of a possible function...

Question 9

Find all the vertical asymptotes of $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$

Accepted Answer

The answer of Find all the vertical asymptotes of \(f...

Question 10

Does $\lim _ { x \rightarrow 2 } \frac { | x - 2 | } { x - 2 }$ exist? Explain why or why not.

Accepted Answer

@#LAT-DLM& does not

Question 11

Given $\lim _ { x \rightarrow 2 } ( x + 3 ) = 5$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

Accepted Answer

A)  ?/2 
B)  ?/3 
C)  ? 
D)  None of these. 
A)  ?/2 
B)  ?/3 
C)  ? 
D)  None of these.

Question 12

Is $f ( x ) = \left\{ \begin{array} { c l } 
x - 2 & \text { if } x < 2 \
- ( x - 2 ) & \text { if } x \geq 2
\end{array} \right.$ continuous at $x = 2 ?$ Explain why or why not.

Accepted Answer

It is cont

Question 13

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 5 x }$

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow 0...

Question 14

Which of the following forms is/are indeterminate?
A. $0 ^ { 1 }$
B. $0 ^ { 0 }$
C. $1 ^ { \infty }$
D. $\infty ^ { 1 }$

Accepted Answer

A)  Only A 
B)  A and D 
C)  B and C 
D)  C and D 
A)  Only A 
B)  A and D 
C)  B and C 
D)  C and D

Question 15

Suppose that $f$ and $g$ are functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 2 } g ( x ) = 3$ Find $\lim _ { x \rightarrow 2 } f ( g ( x ) )$ , if possible. Otherwise, explain why not.?

Accepted Answer

Since @#LAT-DLM& is not contin

Question 16

Is $f ( x ) = \left\{ \begin{array} { c l } 
x - 2 & \text { if } x < 2 \
- ( x - 2 ) & \text { if } x \geq 2
\end{array} \right.$ continuous at $x = 0 ?$ Explain why or why not.

Accepted Answer

It is continuous at

Question 17

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 3 \cos x }$

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow 0...

Question 18

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { \sqrt { 9 x ^ { 2 } - 5 } }$

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow +...

Question 19

Find $\lim _ { x \rightarrow 0 } \frac { 5 x } { \sin 3 x }$

Accepted Answer

A)  3/5 
B)  5/3 
C)  1 
D)  Does not exist 
A)  3/5 
B)  5/3 
C)  1 
D)  Does not exist

Question 20

$f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } + 1 & \text { if } x < - 1 \
3 & \text { if } - 1 \leq x < 2 \
2 x - 1 & \text { if } x \geq 2
\end{array} \right.$ Find $\lim _ { x \rightarrow - 1 } f ( x )$ if it exists.

Accepted Answer

A)  2 
B)  Does not exist 
C)  3 
D)  0 
A)  2 
B)  Does not exist 
C)  3 
D)  0

Is $f ( x ) = \sqrt { x }$ continuous at $x = - 1 ?$ Explain why or why not.

$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 1 & \text { if } x < - 1 \\3 & \text { if } - 1 \leq x < 2 \\2 x - 1 & \text { if } x \geq 2\end{array} \right.$ Find $\lim _ { x \rightarrow 2 } f ( x )$ if it exists.

$f ( x ) = \left\{ \begin{array} { c } x ^ { 2 } \text { if } x < 3 \\2 x + 2 \text { if } x \geq 3\end{array} \right.$ Find $\lim _ { x \rightarrow 3 } f ( x )$ if it exists. Otherwise, explain by one-sided limits.

Use the Intermediate Value Theorem to find a $c \in ( - \infty , \infty )$ for $f ( x ) = x ^ { 3 } + 2$ on the interval [-2, 3] such that $f ( c ) = 1$

Find $\lim _ { x \rightarrow + \infty } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }$

Find $\lim _ { x \rightarrow - \infty } \frac { 1 - 2 x } { \sqrt { 4 x ^ { 2 } - 5 } }$

Given $f ( x ) = \frac { 4 x } { x ^ { 2 } + 3 x }$ , find all the $X$ values where $f$ is not continuous.

Find the equation of a possible function $f$ with $\lim f ( x ) = 0$ and $\lim _ { x \rightarrow + \infty } f ( x ) = 0$ not defined at $x = 3$ , and $f ( 0 ) = - 1$

Find all the vertical asymptotes of $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$

Does $\lim _ { x \rightarrow 2 } \frac { | x - 2 | } { x - 2 }$ exist? Explain why or why not.

Given $\lim _ { x \rightarrow 2 } ( x + 3 ) = 5$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 2 ?$ Explain why or why not.

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 5 x }$

Which of the following forms is/are indeterminate? A. $0 ^ { 1 }$ B. $0 ^ { 0 }$ C. $1 ^ { \infty }$ D. $\infty ^ { 1 }$

Suppose that $f$ and $g$ are functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 2 } g ( x ) = 3$ Find $\lim _ { x \rightarrow 2 } f ( g ( x ) )$ , if possible. Otherwise, explain why not.?

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 0 ?$ Explain why or why not.

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 3 \cos x }$

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { \sqrt { 9 x ^ { 2 } - 5 } }$

Find $\lim _ { x \rightarrow 0 } \frac { 5 x } { \sin 3 x }$

$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 1 & \text { if } x < - 1 \\3 & \text { if } - 1 \leq x < 2 \\2 x - 1 & \text { if } x \geq 2\end{array} \right.$ Find $\lim _ { x \rightarrow - 1 } f ( x )$ if it exists.

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 1: Limits

Is f(x)=xf ( x ) = \sqrt { x }f(x)=x​ continuous at x=−1?x = - 1 ?x=−1? Explain why or why not.

Use the Intermediate Value Theorem to find a c∈(−∞,∞)c \in ( - \infty , \infty )c∈(−∞,∞) for f(x)=x3+2f ( x ) = x ^ { 3 } + 2f(x)=x3+2 on the interval [-2, 3] such that f(c)=1f ( c ) = 1f(c)=1

Find lim⁡x→+∞tan⁡−1(x)3x\lim _ { x \rightarrow + \infty } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }limx→+∞​3xtan−1(x​)​

Find lim⁡x→−∞1−2x4x2−5\lim _ { x \rightarrow - \infty } \frac { 1 - 2 x } { \sqrt { 4 x ^ { 2 } - 5 } }limx→−∞​4x2−5​1−2x​

Given f(x)=4xx2+3xf ( x ) = \frac { 4 x } { x ^ { 2 } + 3 x }f(x)=x2+3x4x​ , find all the XXX values where fff is not continuous.

Find the equation of a possible function fff with lim⁡f(x)=0\lim f ( x ) = 0limf(x)=0 and lim⁡x→+∞f(x)=0\lim _ { x \rightarrow + \infty } f ( x ) = 0limx→+∞​f(x)=0 not defined at x=3x = 3x=3 , and f(0)=−1f ( 0 ) = - 1f(0)=−1

Find all the vertical asymptotes of f(x)=x−2x2−4f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }f(x)=x2−4x−2​

Does lim⁡x→2∣x−2∣x−2\lim _ { x \rightarrow 2 } \frac { | x - 2 | } { x - 2 }limx→2​x−2∣x−2∣​ exist? Explain why or why not.

Given lim⁡x→2(x+3)=5\lim _ { x \rightarrow 2 } ( x + 3 ) = 5limx→2​(x+3)=5 find a δ>0\delta > 0δ>0 in terms of an ε>0\varepsilon > 0ε>0 so that if 0<∣x−c∣<δ0 < | x - c | < \delta0<∣x−c∣<δ then ∣f(x)−L∣<ε| f ( x ) - L | < \varepsilon∣f(x)−L∣<ε

Is f(x)={x−2 if x<2−(x−2) if x≥2f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.f(x)={x−2−(x−2)​ if x<2 if x≥2​ continuous at x=2?x = 2 ?x=2? Explain why or why not.

Find lim⁡x→0cos⁡x−15x\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 5 x }limx→0​5xcosx−1​

Which of the following forms is/are indeterminate? A. 010 ^ { 1 }01 B. 000 ^ { 0 }00 C. 1∞1 ^ { \infty }1∞ D. ∞1\infty ^ { 1 }∞1

Is f(x)={x−2 if x<2−(x−2) if x≥2f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.f(x)={x−2−(x−2)​ if x<2 if x≥2​ continuous at x=0?x = 0 ?x=0? Explain why or why not.

Find lim⁡x→0cos⁡x−13cos⁡x\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 3 \cos x }limx→0​3cosxcosx−1​

Find lim⁡x→+∞1−2x9x2−5\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { \sqrt { 9 x ^ { 2 } - 5 } }limx→+∞​9x2−5​1−2x​

Find lim⁡x→05xsin⁡3x\lim _ { x \rightarrow 0 } \frac { 5 x } { \sin 3 x }limx→0​sin3x5x​

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Is $f ( x ) = \sqrt { x }$ continuous at $x = - 1 ?$ Explain why or why not.

Use the Intermediate Value Theorem to find a $c \in ( - \infty , \infty )$ for $f ( x ) = x ^ { 3 } + 2$ on the interval [-2, 3] such that $f ( c ) = 1$

Find $\lim _ { x \rightarrow + \infty } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }$

Find $\lim _ { x \rightarrow - \infty } \frac { 1 - 2 x } { \sqrt { 4 x ^ { 2 } - 5 } }$

Given $f ( x ) = \frac { 4 x } { x ^ { 2 } + 3 x }$ , find all the $X$ values where $f$ is not continuous.

Find the equation of a possible function $f$ with $\lim f ( x ) = 0$ and $\lim _ { x \rightarrow + \infty } f ( x ) = 0$ not defined at $x = 3$ , and $f ( 0 ) = - 1$

Find all the vertical asymptotes of $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$

Does $\lim _ { x \rightarrow 2 } \frac { | x - 2 | } { x - 2 }$ exist? Explain why or why not.

Given $\lim _ { x \rightarrow 2 } ( x + 3 ) = 5$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 2 ?$ Explain why or why not.

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 5 x }$

Which of the following forms is/are indeterminate? A. $0 ^ { 1 }$ B. $0 ^ { 0 }$ C. $1 ^ { \infty }$ D. $\infty ^ { 1 }$

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 0 ?$ Explain why or why not.

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 3 \cos x }$

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { \sqrt { 9 x ^ { 2 } - 5 } }$

Find $\lim _ { x \rightarrow 0 } \frac { 5 x } { \sin 3 x }$