Question 1

Find the slope of the graph of $f$ at the given point.  $f ( x ) = x ^ { 2 } - 1$

Accepted Answer

A)  $5$ 
B)  $24$ 
C)  $10$ 
D)  $- 10$ 
E)  $- 5$ 
A)  $5$ 
B)  $24$ 
C)  $10$ 
D)  $- 10$ 
E)  $- 5$

Question 2

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ .Use the graph to approximate the slope of the tangent line.   $f ( x ) = x ^ { 2 } - 4 x + 4$

Accepted Answer

A)     Slope at $( - 1,2 ) \approx 2$ 
B)     Slope at $( 1,0 ) \approx 0$ 
C)     Slope at $( 1,1 ) \approx - 2$ 
D)      Slope at $( 1,0 ) \approx 2$ 
E)      Slope at $( 2,0 ) \approx - 2$ 
A)     Slope at $( - 1,2 ) \approx 2$ 
B)     Slope at $( 1,0 ) \approx 0$ 
C)     Slope at $( 1,1 ) \approx - 2$ 
D)      Slope at $( 1,0 ) \approx 2$ 
E)      Slope at $( 2,0 ) \approx - 2$

Question 3

Evaluate the sum using the summation formula and property. 
 $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$

Accepted Answer

A) 1,465 
B) 1,565 
C) 1,165 
D) 1,365 
E) 1,265 
A) 1,465 
B) 1,565 
C) 1,165 
D) 1,365 
E) 1,265

Question 4

Use the graph to find $$\lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 }$$ .

Accepted Answer

A)  $\infty$ 
B) 12 
C) 0 
D) 6 
E) limit does not exist 
A)  $\infty$ 
B) 12 
C) 0 
D) 6 
E) limit does not exist

Question 5

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sin ( 2 x ) } { 2 x }$

Accepted Answer

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

Question 6

Given that $\lim _ { x \rightarrow c } f ( x ) = 7 , \text { and } \lim _ { x \rightarrow c } g ( x ) = 3$ ,find $\lim_{x \rightarrow c} h ( x ) = [ f ( x ) + g ( x ) ] ^ { 2 }$ .

Accepted Answer

A) 42 
B) 100 
C) 63 
D) limit does not exist 
E) 147 
A) 42 
B) 100 
C) 63 
D) limit does not exist 
E) 147

Question 7

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 9 x ^ { 2 } - 1 } { 5 x ^ { 2 } - 5 } \right)$

Accepted Answer

A)  $\frac { 5 } { 9 }$ 
B)  $\frac { 9 } { 5 }$ 
C)  $- \frac { 5 } { 9 }$ 
D)  $- \frac { 9 } { 5 }$ 
E) does not exist 
A)  $\frac { 5 } { 9 }$ 
B)  $\frac { 9 } { 5 }$ 
C)  $- \frac { 5 } { 9 }$ 
D)  $- \frac { 9 } { 5 }$ 
E) does not exist

Question 8

Find $\lim _ { x \rightarrow 2 } g ( x )$ .   $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$

Accepted Answer

A)  $\lim _ { x \rightarrow 2 } g ( x )$ = 
B)  $\lim _ { x \rightarrow 2 } g ( x )$ = $\frac { 1 } { 2 }$ 
C)  $\lim _ { x \rightarrow 2 } g ( x )$ = 4 
D)  $\lim _ { x \rightarrow 2 } g ( x )$ = $\frac { 1 } { 4 }$ 
E)  $\lim _ { x \rightarrow 2 } g ( x )$ = 2 
A)  $\lim _ { x \rightarrow 2 } g ( x )$ = 
B)  $\lim _ { x \rightarrow 2 } g ( x )$ = $\frac { 1 } { 2 }$ 
C)  $\lim _ { x \rightarrow 2 } g ( x )$ = 4 
D)  $\lim _ { x \rightarrow 2 } g ( x )$ = $\frac { 1 } { 4 }$ 
E)  $\lim _ { x \rightarrow 2 } g ( x )$ = 2

Question 9

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = - 2 x + 5&[0,2]\
\end{array}$

Accepted Answer

A) A = 7 square units 
B) A = 8 square units 
C) A = 5square units 
D) A = 4 square units 
E) A = 6 square units 
A) A = 7 square units 
B) A = 8 square units 
C) A = 5square units 
D) A = 4 square units 
E) A = 6 square units

Question 10

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 8 x - x ^ { 3 } &[0,2]\
\end{array}$

Accepted Answer

A) A = 10 square units 
B) A = 8 square units 
C) A = 6 square units 
D) A = 12 square units 
E) A = 14 square units 
A) A = 10 square units 
B) A = 8 square units 
C) A = 6 square units 
D) A = 12 square units 
E) A = 14 square units

Question 11

Find the limit by direct substitution.   $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$

Accepted Answer

A)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\infty$ 
B)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $- \frac { 16 } { 17 }$ 
C)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\frac { 13 } { 17 }$ 
D)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $- \frac { 17 } { 13 }$ 
E)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\frac { 17 } { 16 }$ 
A)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\infty$ 
B)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $- \frac { 16 } { 17 }$ 
C)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\frac { 13 } { 17 }$ 
D)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $- \frac { 17 } { 13 }$ 
E)  $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$ = $\frac { 17 } { 16 }$

Question 12

Rewrite the sum as a rational function S(n).  $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

Accepted Answer

A)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 2 } }$ 
B)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 4 } }$ 
C)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 5 } }$ 
D)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 3 } }$ 
E)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n }$ 
A)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 2 } }$ 
B)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 4 } }$ 
C)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 5 } }$ 
D)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 3 } }$ 
E)  $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n }$

Question 13

Use the given information to evaluate the limit.   $\lim  _ { x \rightarrow c }f ( x ) = 12$ , $\lim _ { x \rightarrow c } g ( x ) = - 4$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$

Accepted Answer

A)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 16 
B)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 8 
C)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 64 
D)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 256 
E)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 4 
A)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 16 
B)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 8 
C)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 64 
D)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 256 
E)  $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$ = 4

Question 14

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.Round your answer to four decimal places.   $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx - 0.7236$    
B)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 1.2236$     
C)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 0.2236$    
D)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 0.7236$    
E)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx - 0.2764$     
A)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx - 0.7236$    
B)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 1.2236$     
C)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 0.2236$    
D)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx 0.7236$    
E)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x } \approx - 0.2764$

Question 15

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result.   $g ( x ) = x ^ { 2 } - 8 x , \quad ( 7 , - 7 )$

Accepted Answer

A)  $m = - 7$ 
B)  $m = 7$ 
C)  $m = - 6$ 
D)  $m = 6$ 
E)  $m = 8$ 
A)  $m = - 7$ 
B)  $m = 7$ 
C)  $m = - 6$ 
D)  $m = 6$ 
E)  $m = 8$

Question 16

Find $\lim _ { x \rightarrow 7 ^ { - } } \frac { x - 7 } { x ^ { 2 } - 49 }$ .

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  $\frac { 1 } { 7 }$ 
D)  $\frac { 2 } { 7 }$ 
E) limit does not exist 
A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  $\frac { 1 } { 7 }$ 
D)  $\frac { 2 } { 7 }$ 
E) limit does not exist

Question 17

Select the graph of the derivative of a function.It is not necessary to find the derivative of the function.   $f ( x ) = x ^ { 5 }$

Accepted Answer

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

Question 18

Find $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$ .

Accepted Answer

A) 32 
B) 4 
C) limit does not exist 
D) 12 
E) 48 
A) 32 
B) 4 
C) limit does not exist 
D) 12 
E) 48

Question 19

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = - \frac { 1 } { 8 }$ 
C)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 9 }$ 
D)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = - \frac { 1 } { 9 }$ 
E)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 10 }$ 
A)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = - \frac { 1 } { 8 }$ 
C)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 9 }$ 
D)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = - \frac { 1 } { 9 }$ 
E)  $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 } = \frac { 1 } { 10 }$

Question 20

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.   $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = - \frac { 1 } { 13 }$   
B)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 13 }$   
C)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 12 }$   
D)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = - \frac { 1 } { 12 }$   
E)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 14 }$   
A)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = - \frac { 1 } { 13 }$   
B)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 13 }$   
C)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 12 }$   
D)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = - \frac { 1 } { 12 }$   
E)  $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 } = \frac { 1 } { 14 }$

Find the slope of the graph of $f$ at the given point. $f ( x ) = x ^ { 2 } - 1$

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ .Use the graph to approximate the slope of the tangent line. $f ( x ) = x ^ { 2 } - 4 x + 4$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 } .$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sin ( 2 x ) } { 2 x }$

Given that $\lim _ { x \rightarrow c } f ( x ) = 7 , \text { and } \lim _ { x \rightarrow c } g ( x ) = 3$ ,find $\lim_{x \rightarrow c} h ( x ) = [ f ( x ) + g ( x ) ] ^ { 2 }$ .

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 9 x ^ { 2 } - 1 } { 5 x ^ { 2 } - 5 } \right)$

Find $\lim _ { x \rightarrow 2 } g ( x )$ . $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=-2x+5 [0,2]

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=8x- [0,2]

Find the limit by direct substitution. $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c }f ( x ) = 12$ , $\lim _ { x \rightarrow c } g ( x ) = - 4$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ] ^ { 2 }$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.Round your answer to four decimal places. $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }$

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. $g ( x ) = x ^ { 2 } - 8 x , \quad ( 7 , - 7 )$

Find $\lim _ { x \rightarrow 7 ^ { - } } \frac { x - 7 } { x ^ { 2 } - 49 }$ .

Select the graph of the derivative of a function.It is not necessary to find the derivative of the function. $f ( x ) = x ^ { 5 }$

Find $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$ .

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 }$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 }$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Filters

Exam 12: Limits and An Introduction To Calculus

Find the slope of the graph of fff at the given point. f(x)=x2−1f ( x ) = x ^ { 2 } - 1f(x)=x2−1

Select a graph of the function and the tangent line at the point (1,f(1))( 1 , f ( 1 ) )(1,f(1)) .Use the graph to approximate the slope of the tangent line. f(x)=x2−4x+4f ( x ) = x ^ { 2 } - 4 x + 4f(x)=x2−4x+4

Evaluate the sum using the summation formula and property. ∑k=135(2k+3)\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )∑k=135​(2k+3)

Use the graph to find lim⁡x→32x2−18x−3\lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 }limx→3​x−32x2−18​ .

Select the correct graph for the following function using a graphing utility. f(x)=sin⁡(2x)2xf ( x ) = \frac { \sin ( 2 x ) } { 2 x }f(x)=2xsin(2x)​

Find the limit (if it exists). lim⁡x→∞(9x2−15x2−5)\lim _ { x \rightarrow \infty } \left( \frac { 9 x ^ { 2 } - 1 } { 5 x ^ { 2 } - 5 } \right)limx→∞​(5x2−59x2−1​)

Find lim⁡x→2g(x)\lim _ { x \rightarrow 2 } g ( x )limx→2​g(x) . g(x)=x2+53x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }g(x)=3x2x2+5​​

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=-2x+5 [0,2]

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=8x- [0,2]

Find the limit by direct substitution. lim⁡x→−4(4xx2+1)\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)limx→−4​(x2+14x​)

Rewrite the sum as a rational function S(n). ∑i=1ni3n3\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }∑i=1n​n3i3​

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.Round your answer to four decimal places. lim⁡x→0−x+5−5x\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }limx→0−​xx+5​−5​​

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. g(x)=x2−8x,(7,−7)g ( x ) = x ^ { 2 } - 8 x , \quad ( 7 , - 7 )g(x)=x2−8x,(7,−7)

Find lim⁡x→7−x−7x2−49\lim _ { x \rightarrow 7 ^ { - } } \frac { x - 7 } { x ^ { 2 } - 49 }limx→7−​x2−49x−7​ .

Select the graph of the derivative of a function.It is not necessary to find the derivative of the function. f(x)=x5f ( x ) = x ^ { 5 }f(x)=x5

Find lim⁡t→4t3−64t−4\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }limt→4​t−4t3−64​ .

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). lim⁡x→4x−4x2−16\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 }limx→4​x2−16x−4​

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. lim⁡x→36+6−xx−36\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 }limx→36+​x−366−x​​

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Filters

Find the slope of the graph of $f$ at the given point. $f ( x ) = x ^ { 2 } - 1$

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ .Use the graph to approximate the slope of the tangent line. $f ( x ) = x ^ { 2 } - 4 x + 4$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 2 x ^ { 2 } - 18 } { x - 3 } .$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sin ( 2 x ) } { 2 x }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 9 x ^ { 2 } - 1 } { 5 x ^ { 2 } - 5 } \right)$

Find $\lim _ { x \rightarrow 2 } g ( x )$ . $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 4 } \left( \frac { 4 x } { x ^ { 2 } + 1 } \right)$

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.Round your answer to four decimal places. $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }$

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. $g ( x ) = x ^ { 2 } - 8 x , \quad ( 7 , - 7 )$

Find $\lim _ { x \rightarrow 7 ^ { - } } \frac { x - 7 } { x ^ { 2 } - 49 }$ .

Select the graph of the derivative of a function.It is not necessary to find the derivative of the function. $f ( x ) = x ^ { 5 }$

Find $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$ .

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 16 }$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 36 ^ { + } } \frac { 6 - \sqrt { x } } { x - 36 }$