Question 1

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $\frac { 6 } { 5 }$ 
C)  $\infty$ 
D)  $\frac { 1 } { 30 }$ 
E)  limit does not exist 
A)  0 
B)  $\frac { 6 } { 5 }$ 
C)  $\infty$ 
D)  $\frac { 1 } { 30 }$ 
E)  limit does not exist

Question 2

Use the graph to determine $ \lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x} $ (if it exists).

Accepted Answer

A)  0 
B)  3 
C)   -3 
D)  6 
E)  limit does not exist 
A)  0 
B)  3 
C)   -3 
D)  6 
E)  limit does not exist

Question 3

Complete the table and use the result to estimate $\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-x-6}$ numerically.

$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -2.1 & -2.01 & -2.001 & -2 & -1.999 & -1.99 & -1.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $ -\frac{1}{5} $ 
B)  5 
C)  $ \infty $ 
D)   -5 
E)  limit does not exist 
A)  $ -\frac{1}{5} $ 
B)  5 
C)  $ \infty $ 
D)   -5 
E)  limit does not exist

Question 4

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$$

Accepted Answer

A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist. 
A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist.

Question 5

$\text { Find the following limit of the sequence as } n \text { approaches infinity, if it exists. }$ $a_{n}=\frac{1}{n}\left(n+\frac{3}{n}\left[\frac{n(n+1)}{5}\right]\right)$

Accepted Answer

A)  $ \frac{3}{5} $ 
B)  $ \frac{13}{5} $ 
C)  $ \frac{8}{5} $ 
D)  $ \frac{9}{5} $ 
E)  The limit does not exist. 
A)  $ \frac{3}{5} $ 
B)  $ \frac{13}{5} $ 
C)  $ \frac{8}{5} $ 
D)  $ \frac{9}{5} $ 
E)  The limit does not exist.

Question 6

Evaluate $ \sum_{j=1}^{15}\left(j^{3}-5 j^{2}\right) $ using the summation formulas and properties.

Accepted Answer

A)  14,400 
B)  13,275 
C)  14,395 
D)  14,325 
E)  8,200 
A)  14,400 
B)  13,275 
C)  14,395 
D)  14,325 
E)  8,200

Question 7

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Accepted Answer

A)  30 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $2 \sqrt { 15 }$ 
E)  limit does not exist 
A)  30 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $2 \sqrt { 15 }$ 
E)  limit does not exist

Question 8

Find $$\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$$
by direct substitution.

Accepted Answer

A)  $- 15$ 
B)  33 
C)  $- 3$ 
D)  $- 29$ 
E)  limit does not exist 
A)  $- 15$ 
B)  33 
C)  $- 3$ 
D)  $- 29$ 
E)  limit does not exist

Question 9

Find the following limit, if it exists. $$\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$$

Accepted Answer

A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist. 
A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist.

Question 10

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists).

Accepted Answer

A)  0 
B)  1 
C)  $- 1$ 
D)  2 
E)  limit does not exist 
A)  0 
B)  1 
C)  $- 1$ 
D)  2 
E)  limit does not exist

Question 11

$\text { Consider the following graph of the function and approximate } \lim _{x \rightarrow 0} \frac{e^{-2 x}-1}{x} \text {, if it exists. }$

Accepted Answer

A)   -1 
B)  -2 
C)  0 
D)  1 
E)  The limit does not exist. 
A)   -1 
B)  -2 
C)  0 
D)  1 
E)  The limit does not exist.

Question 12

$\text { Rewrite } \sum_{i=1}^{n} \frac{9 i^{3}}{n^{5}} \text { as a rational function } S(n) \text { and find } \lim _{n \rightarrow \infty} S(n) \text {. }$

Accepted Answer

A) $S(n)=\frac{9(n+1)^{2}}{4 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{9}{4}$ 
B) $S(n)=\frac{9 n^{2}(n+1)^{2}}{4}, \lim _{n \rightarrow \infty} S(n)=9$ 
C) $S(n)=\frac{n^{2}(n+1)^{2}}{36}, \lim _{n \rightarrow \infty} S(n)=0$ 
D) $S(n)=\frac{9(n+1)^{2}}{4 n^{3}}, \lim _{n \rightarrow \infty} S(n)=0$ 
E)  $S(n)=\frac{9(n+1)^{2}}{4 n^{3}}, \text { the limit does not exist }$ 
A) $S(n)=\frac{9(n+1)^{2}}{4 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{9}{4}$ 
B) $S(n)=\frac{9 n^{2}(n+1)^{2}}{4}, \lim _{n \rightarrow \infty} S(n)=9$ 
C) $S(n)=\frac{n^{2}(n+1)^{2}}{36}, \lim _{n \rightarrow \infty} S(n)=0$ 
D) $S(n)=\frac{9(n+1)^{2}}{4 n^{3}}, \lim _{n \rightarrow \infty} S(n)=0$ 
E)  $S(n)=\frac{9(n+1)^{2}}{4 n^{3}}, \text { the limit does not exist }$

Question 13

Find
$$\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$$
by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 7 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 3 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 10 }$ 
A)  $\frac { \sqrt { 7 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 3 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 10 }$

Question 14

Find the slope of the graph of the following function at the point $( - 1,1 )$.
$$3 x ^ { 2 } - 2$$

Accepted Answer

A)  4 
B)  $- 6$ 
C)  6 
D)  1 
E)  $- 8$ 
A)  4 
B)  $- 6$ 
C)  6 
D)  1 
E)  $- 8$

Question 15

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$-axis on the interval $[ 0,5 ]$.

Accepted Answer

A)  185 
B)  110 
C)  65 
D)  75 
E)  82 
A)  185 
B)  110 
C)  65 
D)  75 
E)  82

Question 16

Use the limit process to find the area of the region between $ f(x)=x^{2}+5 $ and the $ x $-axis on the interval $ [0,6] $.

Accepted Answer

A)  246 
B)  72 
C)  221 
D)  102 
E)  77 
A)  246 
B)  72 
C)  221 
D)  102 
E)  77

Question 17

$\text { Graph } f(x)=\left\{\begin{array}{ll}
4 x+3 & x<0 \
x+3 & x \geq 0
\end{array} \text { and find the limit of } f(x) \text { as } x \text { approaches } 0 .\right.$

Accepted Answer

A)  15 
B)  3 
C)  12 
D)  -3 
E)  limit does not exist 
A)  15 
B)  3 
C)  12 
D)  -3 
E)  limit does not exist

Question 18

The cost function for a certain graphing calculator is given by $C = 11.35 x + 53,000$ where $C$ is in dollars and $x$ is the number of calculators produced.
a. Write a model for the average cost per unit produced.
b. Find the average cost per unit when $x = 900$.
c. Determine the limit of the average cost function as $x$ approaches $\infty$.

Accepted Answer

The answer of The cost function for a certain graphing...

Question 19

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$.
$$g ( x ) = - 7 - 4 x$$

Accepted Answer

A)  $- 11$ 
B)  $- 27$ 
C)  $- 3$ 
D)  $- 4$ 
E)  5 
A)  $- 11$ 
B)  $- 27$ 
C)  $- 3$ 
D)  $- 4$ 
E)  5

Question 20

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$, if it exists.

Accepted Answer

A)  1 
B)  4 
C)  8 
D)  $- 1$ 
E)  The limit does not exist. 
A)  1 
B)  4 
C)  8 
D)  $- 1$ 
E)  The limit does not exist.

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Use the graph to determine $\lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x}$ (if it exists). $Use the graph to determine \lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x} (if it exists).$

Complete the table and use the result to estimate $\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-x-6}$ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

$\text { Find the following limit of the sequence as } n \text { approaches infinity, if it exists. }$ $a_{n}=\frac{1}{n}\left(n+\frac{3}{n}\left[\frac{n(n+1)}{5}\right]\right)$

Evaluate $\sum_{j=1}^{15}\left(j^{3}-5 j^{2}\right)$ using the summation formulas and properties.

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$ by direct substitution.

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x } (if it exists).$

$\text { Consider the following graph of the function and approximate } \lim _{x \rightarrow 0} \frac{e^{-2 x}-1}{x} \text {, if it exists. }$ $\text { Consider the following graph of the function and approximate } \lim _{x \rightarrow 0} \frac{e^{-2 x}-1}{x} \text {, if it exists. }$

$\text { Rewrite } \sum_{i=1}^{n} \frac{9 i^{3}}{n^{5}} \text { as a rational function } S(n) \text { and find } \lim _{n \rightarrow \infty} S(n) \text {. }$

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$ by direct substitution.

Find the slope of the graph of the following function at the point $( - 1,1 )$ . $3 x ^ { 2 } - 2$

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$ -axis on the interval $[ 0,5 ]$ .

Use the limit process to find the area of the region between $f(x)=x^{2}+5$ and the $x$ -axis on the interval $[0,6]$ .

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$ . $g ( x ) = - 7 - 4 x$

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$ , if it exists. $Consider the graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x } , if it exists.$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Exam 11: Calculus Practice Problems

Find lim⁡x→∞65x\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }limx→∞​5x6​ (if it exists).

Use the graph to determine lim⁡x→0x2−3xx \lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x} limx→0​xx2−3x​ (if it exists).

Complete the table and use the result to estimate lim⁡x→−2x+2x2−x−6\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-x-6}limx→−2​x2−x−6x+2​ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

Find the following limit, if it exists. lim⁡x→∞5x8\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }limx→∞​x85​

Evaluate ∑j=115(j3−5j2) \sum_{j=1}^{15}\left(j^{3}-5 j^{2}\right) ∑j=115​(j3−5j2) using the summation formulas and properties.

Find lim⁡x→0−xx+15−15\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }limx→0−​x+15​−15​x​

Find lim⁡x→−3(13x3−2x)\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)limx→−3​(31​x3−2x) by direct substitution.

Find the following limit, if it exists. lim⁡x→∞4x2+3−2−x2\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }limx→∞​−2−x24x2+3​

Use the graph to determine lim⁡x→0x2−xx\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }limx→0​xx2−x​ (if it exists).

Find lim⁡x→7x+3x−3\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }limx→7​x−3x+3​​ by direct substitution.

Find the slope of the graph of the following function at the point (−1,1)( - 1,1 )(−1,1) . 3x2−23 x ^ { 2 } - 23x2−2

Use the limit process to find the area of the region between f(x)=6x+7f ( x ) = 6 x + 7f(x)=6x+7 and the xxx -axis on the interval [0,5][ 0,5 ][0,5] .

Use the limit process to find the area of the region between f(x)=x2+5 f(x)=x^{2}+5 f(x)=x2+5 and the x x x -axis on the interval [0,6] [0,6] [0,6] .

Use the limit process to find the slope of the graph of the following function at the point (5,−27)( 5 , - 27 )(5,−27) . g(x)=−7−4xg ( x ) = - 7 - 4 xg(x)=−7−4x

Consider the graph of the function and approximate lim⁡x→0sin⁡(8x)x\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }limx→0​xsin(8x)​ , if it exists.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Use the graph to determine $\lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x}$ (if it exists). $Use the graph to determine \lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x} (if it exists).$

Complete the table and use the result to estimate $\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-x-6}$ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

Evaluate $\sum_{j=1}^{15}\left(j^{3}-5 j^{2}\right)$ using the summation formulas and properties.

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$ by direct substitution.

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x } (if it exists).$

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$ by direct substitution.

Find the slope of the graph of the following function at the point $( - 1,1 )$ . $3 x ^ { 2 } - 2$

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$ -axis on the interval $[ 0,5 ]$ .

Use the limit process to find the area of the region between $f(x)=x^{2}+5$ and the $x$ -axis on the interval $[0,6]$ .

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$ . $g ( x ) = - 7 - 4 x$

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$ , if it exists. $Consider the graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x } , if it exists.$