Question 1

Find
$$\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$$
by direct substitution.

Accepted Answer

A)  $\frac { 6 } { \sqrt { 10 } }$ 
B)  $4 \sqrt { 6 }$ 
C)  $24 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 24 } { \sqrt { 10 } }$ 
A)  $\frac { 6 } { \sqrt { 10 } }$ 
B)  $4 \sqrt { 6 }$ 
C)  $24 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 24 } { \sqrt { 10 } }$

Question 2

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$, if it exists.

Accepted Answer

A)  $- 1$ 
B)  $- 4$ 
C)  0 
D)  1 
E)  The limit does not exist. 
A)  $- 1$ 
B)  $- 4$ 
C)  0 
D)  1 
E)  The limit does not exist.

Question 3

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point

Question 4

Determine any points on the graph of the following function at which the tangent line is horizontal. $$f ( x ) = x ^ { 2 } + 6 x - 8$$

Accepted Answer

A)  $( 3,19 )$ 
B)  $( 3,0 )$ 
C)  $( - 3,0 )$ 
D)  $( 1 , - 1 )$ 
E)  $( - 3 , - 17 )$ 
A)  $( 3,19 )$ 
B)  $( 3,0 )$ 
C)  $( - 3,0 )$ 
D)  $( 1 , - 1 )$ 
E)  $( - 3 , - 17 )$

Question 5

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$, if it exists.

Accepted Answer

A)  $- 1$ 
B)  $- 4$ 
C)  0 
D)  1 
E)  The limit does not exist. 
A)  $- 1$ 
B)  $- 4$ 
C)  0 
D)  1 
E)  The limit does not exist.

Question 6

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

Accepted Answer

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

Question 7

Use the limit process to find the slope of the graph of $ 6 x-4 x^{2} $ at $ (7,-154) $.

Accepted Answer

A)   -2 
B)  62 
C)  34 
D)  56 
E)  -50 
A)   -2 
B)  62 
C)  34 
D)  56 
E)  -50

Question 8

Find an equation of the tangent line to the graph of the following function at the point $ (-1,-5) $
$ -3 x^{2}-2 $

Accepted Answer

A)  $ y=6 x-11 $ 
B)  $ y=6 x+11 $ 
C)  $ y=6 x+1 $ 
D)  $ y=30 x-25 $ 
E)  $ y=30 x+25 $ 
A)  $ y=6 x-11 $ 
B)  $ y=6 x+11 $ 
C)  $ y=6 x+1 $ 
D)  $ y=30 x-25 $ 
E)  $ y=30 x+25 $

Question 9

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Accepted Answer

A)  $- \frac { 5 } { 49 }$ 
B)  6 
C)  $- 6$ 
D)  $\infty$ 
E)  limit does not exist 
A)  $- \frac { 5 } { 49 }$ 
B)  6 
C)  $- 6$ 
D)  $\infty$ 
E)  limit does not exist

Question 10

Use asymptotes to match  $f(x)=\frac{4 x^{2}}{x^{2}+1}$ with its graph.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 11

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 12

Find
$$\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$$
by direct substitution.

Accepted Answer

A)  $\frac { 8 } { \sqrt { 10 } }$ 
B)  $10 \sqrt { 2 }$ 
C)  $40 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 40 } { \sqrt { 10 } }$ 
A)  $\frac { 8 } { \sqrt { 10 } }$ 
B)  $10 \sqrt { 2 }$ 
C)  $40 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 40 } { \sqrt { 10 } }$

Question 13

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }$$

Accepted Answer

A)  $- 3$ 
B)  9 
C)  7 
D)  $- 11$ 
E)  The limit does not exist. 
A)  $- 3$ 
B)  9 
C)  7 
D)  $- 11$ 
E)  The limit does not exist.

Question 14

Find the following limit.
$$\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$$

Accepted Answer

A)  52 
B)  36 
C)  37 
D)  $- 23$ 
E)  $- 35$ 
A)  52 
B)  36 
C)  37 
D)  $- 23$ 
E)  $- 35$

Question 15

Using the summation formulas and properties, evaluate the following expression.
$$\sum _ { i = 1 } ^ { 60 } 4$$

Accepted Answer

A)  7320 
B)  1830 
C)  236 
D)  240 
E)  64 
A)  7320 
B)  1830 
C)  236 
D)  240 
E)  64

Question 16

Use the limit process to find the slope of the graph of $ \sqrt{x+21} $ at $ (4,5) $.

Accepted Answer

A)  $ \frac{1}{5} $ 
B)  5 
C)  $ \infty $ 
D)  $ \frac{1}{10} $ 
E)  the slope is undefined at this point 
A)  $ \frac{1}{5} $ 
B)  5 
C)  $ \infty $ 
D)  $ \frac{1}{10} $ 
E)  the slope is undefined at this point

Question 17

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1 ).

Accepted Answer

A)  $\frac { 5 } { 3 }$ 
B)  111 
C)  79 
D)  0 
E)  the sequence diverges 
A)  $\frac { 5 } { 3 }$ 
B)  111 
C)  79 
D)  0 
E)  the sequence diverges

Question 18

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 2 + 6 x$.

Accepted Answer

A)  $- 6$ 
B)  $- 4$ 
C)  6 
D)  2 
E)  limit does not exist 
A)  $- 6$ 
B)  $- 4$ 
C)  6 
D)  2 
E)  limit does not exist

Question 19

Complete the table and use the result to estimate    $\lim _{x \rightarrow-3} \frac{x+3}{x^{2}-x-12}$  numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -3.1 & -3.01 & -3.001 & -3 & -2.999 & -2.99 & -2.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

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

Question 20

Use the function below and its derivative to determine any points on the graph of $ f $ at which the tangent line is horizontal.
$f(x)=3 x^{4}-6 x^{2}, \quad f^{\prime}(x)=12 x^{3}-12 x$

Accepted Answer

A)  $ (-1,-3),(0,0),(1,-3) $ 
B)  $ (-1,3),(1,12) $ 
C)  $ (-1,-3),(1,-3) $ 
D)  $ (0,0),(1,0) $ 
E)  $ (0,0),(1,-3) $ 
A)  $ (-1,-3),(0,0),(1,-3) $ 
B)  $ (-1,3),(1,12) $ 
C)  $ (-1,-3),(1,-3) $ 
D)  $ (0,0),(1,0) $ 
E)  $ (0,0),(1,-3) $

Find $\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$ by direct substitution.

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

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

Use the limit process to find the slope of the graph of $6 x-4 x^{2}$ at $(7,-154)$ .

Find an equation of the tangent line to the graph of the following function at the point $(-1,-5)$ $-3 x^{2}-2$

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Use asymptotes to match $f(x)=\frac{4 x^{2}}{x^{2}+1}$ with its graph.

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

Find $\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$ by direct substitution.

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }$

Find the following limit. $\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 60 } 4$

Use the limit process to find the slope of the graph of $\sqrt{x+21}$ at $(4,5)$ .

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1 ).

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 2 + 6 x$ .

Complete the table and use the result to estimate $\lim _{x \rightarrow-3} \frac{x+3}{x^{2}-x-12}$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f(x)=3 x^{4}-6 x^{2}, \quad f^{\prime}(x)=12 x^{3}-12 x$

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

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Exam 11: Calculus Practice Problems

Find lim⁡x→64xx+4\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }limx→6​x+4​4x​ by direct substitution.

Consider the following graph of the function and approximate lim⁡x→0e−4x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }limx→0​xe−4x−1​ , if it exists.

Use the limit process to find the slope of the graph of x+4\sqrt { x + 4 }x+4​ at (5,3)( 5,3 )(5,3) .

Determine any points on the graph of the following function at which the tangent line is horizontal. f(x)=x2+6x−8f ( x ) = x ^ { 2 } + 6 x - 8f(x)=x2+6x−8

Consider the following graph of the function and approximate lim⁡x→0e−4x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }limx→0​xe−4x−1​ , if it exists.

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

Use the limit process to find the slope of the graph of 6x−4x2 6 x-4 x^{2} 6x−4x2 at (7,−154) (7,-154) (7,−154) .

Find an equation of the tangent line to the graph of the following function at the point (−1,−5) (-1,-5) (−1,−5) −3x2−2 -3 x^{2}-2 −3x2−2

Find lim⁡x→−∞[x(7+x)2−6]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]limx→−∞​[(7+x)2x​−6] (if it exists).

Use asymptotes to match f(x)=4x2x2+1f(x)=\frac{4 x^{2}}{x^{2}+1}f(x)=x2+14x2​ with its graph.

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​ .

Find lim⁡x→85xx+2\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }limx→8​x+2​5x​ by direct substitution.

Find the following limit, if it exists. lim⁡x→−39−3x−2x23+x\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }limx→−3​3+x9−3x−2x2​

Find the following limit. lim⁡x→−43x2+3x+1\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1limx→−4​3x2+3x+1

Using the summation formulas and properties, evaluate the following expression. ∑i=1604\sum _ { i = 1 } ^ { 60 } 4∑i=160​4

Use the limit process to find the slope of the graph of x+21 \sqrt{x+21} x+21​ at (4,5) (4,5) (4,5) .

Use the first six terms to predict the limit of the sequence an=5n3+8n+3a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }an​=n+35n3+8​ (assume nnn begins with 1 ).

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)=2+6xf ( x ) = 2 + 6 xf(x)=2+6x .

Complete the table and use the result to estimate lim⁡x→−3x+3x2−x−12\lim _{x \rightarrow-3} \frac{x+3}{x^{2}-x-12}limx→−3​x2−x−12x+3​ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of f f f at which the tangent line is horizontal. f(x)=3x4−6x2,f′(x)=12x3−12xf(x)=3 x^{4}-6 x^{2}, \quad f^{\prime}(x)=12 x^{3}-12 xf(x)=3x4−6x2,f′(x)=12x3−12x

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 $\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$ by direct substitution.

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

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

Use the limit process to find the slope of the graph of $6 x-4 x^{2}$ at $(7,-154)$ .

Find an equation of the tangent line to the graph of the following function at the point $(-1,-5)$ $-3 x^{2}-2$

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Use asymptotes to match $f(x)=\frac{4 x^{2}}{x^{2}+1}$ with its graph.

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

Find $\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$ by direct substitution.

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }$

Find the following limit. $\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 60 } 4$

Use the limit process to find the slope of the graph of $\sqrt{x+21}$ at $(4,5)$ .

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1 ).

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 2 + 6 x$ .

Complete the table and use the result to estimate $\lim _{x \rightarrow-3} \frac{x+3}{x^{2}-x-12}$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f(x)=3 x^{4}-6 x^{2}, \quad f^{\prime}(x)=12 x^{3}-12 x$