Question 1

Estimate the following limit numerically, if it exists. $$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$$

Accepted Answer

A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist.

Question 2

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$.

Accepted Answer

A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist

Question 3

$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$

Accepted Answer

A)  $ \frac{1}{7 \sqrt{x-7}} $ 
B)  $ \frac{1}{2 \sqrt{x-7}} $ 
C)  $ \frac{\sqrt{x-7}}{2} $ 
D)  $ \sqrt{x-7} $ 
E)  limit does not exist 
A)  $ \frac{1}{7 \sqrt{x-7}} $ 
B)  $ \frac{1}{2 \sqrt{x-7}} $ 
C)  $ \frac{\sqrt{x-7}}{2} $ 
D)  $ \sqrt{x-7} $ 
E)  limit does not exist

Question 4

Use the limit process to find the slope of the graph of the following function at the point $ (3,-19) $.
$g(x)=-10-3 x$

Accepted Answer

A)  $ -13 $ 
B)  $ -19 $ 
C)  $ -7 $ 
D)  $ -3 $ 
E)  $ 3$ 
A)  $ -13 $ 
B)  $ -19 $ 
C)  $ -7 $ 
D)  $ -3 $ 
E)  $ 3$

Question 5

$\text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. }$

Accepted Answer

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

Question 6

Find $ \lim _{x \rightarrow 13^{-}} \frac{x-13}{x^{2}-169} $.

Accepted Answer

A)  $ \frac{1}{26} $ 
B)  $ \frac{1}{13} $ 
C)  $ \frac{13}{2} $ 
D)  $ \frac{2}{13} $ 
E)  limit does not exist 
A)  $ \frac{1}{26} $ 
B)  $ \frac{1}{13} $ 
C)  $ \frac{13}{2} $ 
D)  $ \frac{2}{13} $ 
E)  limit does not exist

Question 7

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$.

Accepted Answer

A)  $8 x + 2$ 
B)  $4 x + 2$ 
C)  $8 x + 5$ 
D)  $4 x + 5$ 
E)  $8 x$ 
A)  $8 x + 2$ 
B)  $4 x + 2$ 
C)  $8 x + 5$ 
D)  $4 x + 5$ 
E)  $8 x$

Question 8

Approximate the area of the region under the function below on the interval $ [0,2] $ using 8 rectangles. Round your answer to two decimals.
$f(x)=\frac{1}{3} x^{4}$

Accepted Answer

A)  $ 5.71 $ square units 
B)  $ 2.86 $ square units 
C)  $ 3.69 $ square units 
D)  $ 8.57 $ square units 
E)  $ 17.13 $ square units 
A)  $ 5.71 $ square units 
B)  $ 2.86 $ square units 
C)  $ 3.69 $ square units 
D)  $ 8.57 $ square units 
E)  $ 17.13 $ square units

Question 9

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$-axis on the interval $[ 1,4 ]$.

Accepted Answer

A)  $\frac { 143 } { 2 }$ 
B)  $\frac { 81 } { 4 }$ 
C)  $\frac { 47 } { 2 }$ 
D)  78 
E)  $\frac { 51 } { 4 }$ 
A)  $\frac { 143 } { 2 }$ 
B)  $\frac { 81 } { 4 }$ 
C)  $\frac { 47 } { 2 }$ 
D)  78 
E)  $\frac { 51 } { 4 }$

Question 10

Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist

Question 11

Complete the table and use the result to estimate
$$\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|c}
\hline x & -8.1 & -8.01 & -8.001 & -8 & -7.999 & -7.99 & -7.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist

Question 12

Estimate the following limit numerically, if it exists. $$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$$

Accepted Answer

A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist.

Question 13

Find the following limit, if it exists. $\lim _{x \rightarrow \infty} \frac{6 x^{2}-4}{-3-x^{2}}$

Accepted Answer

A)  $ -\frac{1}{2} $ 
B)  -6 
C)  6 
D)  -4 
E)  The limit does not exist. 
A)  $ -\frac{1}{2} $ 
B)  -6 
C)  6 
D)  -4 
E)  The limit does not exist.

Question 14

A union contract guarantees a $16 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,500$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is given by
$$f ( t ) = \left\{ \begin{array} { l l } 
31.50 , & 0 < t \leq 1 \
36.54 , & 1 < t \leq 2 \
42.39 , & 2 < t \leq 3
\end{array} \right.$$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.

Accepted Answer

A)  $31.50$ 
B)  $36.54$ 
C)  $1.00$ 
D)  $42.39$ 
E)  The limit does not exist. 
A)  $31.50$ 
B)  $36.54$ 
C)  $1.00$ 
D)  $42.39$ 
E)  The limit does not exist.

Question 15

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$.

Accepted Answer

A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist 
A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist

Question 16

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$, if it exists.

Accepted Answer

A)  1 
B)  $- 10$ 
C)  10 
D)  $- 1$ 
E)  The limit does not exist. 
A)  1 
B)  $- 10$ 
C)  10 
D)  $- 1$ 
E)  The limit does not exist.

Question 17

Find $ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $ for $ f(x)=4-6 x $.

Accepted Answer

A)  6 
B)  10 
C)  -6 
D)  4 
E)  limit does not exist 
A)  6 
B)  10 
C)  -6 
D)  4 
E)  limit does not exist

Question 18

Use the figure below to approximate the slope of the curve at the point $( x , y )$.

Accepted Answer

A)  1 
B)  $\frac { 1 } { 2 }$ 
C)  $- 1$ 
D)  0 
E)  The slope is undefined. 
A)  1 
B)  $\frac { 1 } { 2 }$ 
C)  $- 1$ 
D)  0 
E)  The slope is undefined.

Question 19

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

Accepted Answer

A)  $ -\frac{3}{2} $ 
B)  0 
C)  1 
D)  $ \frac{3}{2} $ 
E)  -1 
A)  $ -\frac{3}{2} $ 
B)  0 
C)  1 
D)  $ \frac{3}{2} $ 
E)  -1

Question 20

Find $ \lim _{x \rightarrow 0^{-}} \frac{x}{\sqrt{x+3}-\sqrt{3}} $

Accepted Answer

A)  6 
B)  $ \frac{3}{2} $ 
C)  $ \frac{\sqrt{3}}{2} $ 
D)  $ 2 \sqrt{3} $ 
E)  limit does not exist 
A)  6 
B)  $ \frac{3}{2} $ 
C)  $ \frac{\sqrt{3}}{2} $ 
D)  $ 2 \sqrt{3} $ 
E)  limit does not exist

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$

Use the limit process to find the slope of the graph of the following function at the point $(3,-19)$ . $g(x)=-10-3 x$

$\text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. }$ $\text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. }$

Find $\lim _{x \rightarrow 13^{-}} \frac{x-13}{x^{2}-169}$ .

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

Approximate the area of the region under the function below on the interval $[0,2]$ using 8 rectangles. Round your answer to two decimals. $f(x)=\frac{1}{3} x^{4}$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$ -axis on the interval $[ 1,4 ]$ .

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Find the following limit, if it exists. $\lim _{x \rightarrow \infty} \frac{6 x^{2}-4}{-3-x^{2}}$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 } , if it exists.$

Find $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=4-6 x$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$ $\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

Find $\lim _{x \rightarrow 0^{-}} \frac{x}{\sqrt{x+3}-\sqrt{3}}$

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

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Exam 11: Calculus Practice Problems

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+6x−7\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }limx→1​x2+6x−7x−1​

Find lim⁡x→15−x−15x2−225\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }limx→15−​x2−225x−15​ .

Find lim⁡h→0f(x+h)−f(x)h for f(x)=x−7. \text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. } Find limh→0​hf(x+h)−f(x)​ for f(x)=x−7​.

Use the limit process to find the slope of the graph of the following function at the point (3,−19) (3,-19) (3,−19) . g(x)=−10−3xg(x)=-10-3 xg(x)=−10−3x

Use the graph below to find lim⁡x→−4∣x+4∣x+4, if it exists. \text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. } Use the graph below to find limx→−4​x+4∣x+4∣​, if it exists.

Find lim⁡x→13−x−13x2−169 \lim _{x \rightarrow 13^{-}} \frac{x-13}{x^{2}-169} limx→13−​x2−169x−13​ .

Find the derivative of f(x)=4x2+2x+3f ( x ) = 4 x ^ { 2 } + 2 x + 3f(x)=4x2+2x+3 .

Approximate the area of the region under the function below on the interval [0,2] [0,2] [0,2] using 8 rectangles. Round your answer to two decimals. f(x)=13x4f(x)=\frac{1}{3} x^{4}f(x)=31​x4

Use the limit process to find the area of the region between f(x)=14(x2+4x)f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)f(x)=41​(x2+4x) and the xxx -axis on the interval [1,4][ 1,4 ][1,4] .

Find lim⁡x→3[g(x)−f(x)]\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]limx→3​[g(x)−f(x)] for f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 and g(x)=x2+95x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }g(x)=5x2x2+9​​ .

Complete the table and use the result to estimate lim⁡x→−8x+8x2+4x−32\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }limx→−8​x2+4x−32x+8​ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+5x−6\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }limx→1​x2+5x−6x−1​

Find the following limit, if it exists. lim⁡x→∞6x2−4−3−x2\lim _{x \rightarrow \infty} \frac{6 x^{2}-4}{-3-x^{2}}limx→∞​−3−x26x2−4​

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​ .

Use the graph below to find lim⁡x→−10∣x+10∣x+10\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }limx→−10​x+10∣x+10∣​ , if it exists.

Find lim⁡h→0f(x+h)−f(x)h \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} limh→0​hf(x+h)−f(x)​ for f(x)=4−6x f(x)=4-6 x f(x)=4−6x .

Use the figure below to approximate the slope of the curve at the point (x,y)( x , y )(x,y) .

Use the figure below to approximate the slope of the curve at the point (x,y). \text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. } Use the figure below to approximate the slope of the curve at the point (x,y).

Find lim⁡x→0−xx+3−3 \lim _{x \rightarrow 0^{-}} \frac{x}{\sqrt{x+3}-\sqrt{3}} limx→0−​x+3​−3​x​

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

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$

Use the limit process to find the slope of the graph of the following function at the point $(3,-19)$ . $g(x)=-10-3 x$

$\text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. }$ $\text { Use the graph below to find } \lim _{x \rightarrow-4} \frac{|x+4|}{x+4} \text {, if it exists. }$

Find $\lim _{x \rightarrow 13^{-}} \frac{x-13}{x^{2}-169}$ .

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

Approximate the area of the region under the function below on the interval $[0,2]$ using 8 rectangles. Round your answer to two decimals. $f(x)=\frac{1}{3} x^{4}$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$ -axis on the interval $[ 1,4 ]$ .

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Find the following limit, if it exists. $\lim _{x \rightarrow \infty} \frac{6 x^{2}-4}{-3-x^{2}}$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 } , if it exists.$

Find $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=4-6 x$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$ $\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

Find $\lim _{x \rightarrow 0^{-}} \frac{x}{\sqrt{x+3}-\sqrt{3}}$