Question 1

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ .   $f ( x ) = \frac { 1 } { x + 5 }$

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 5 } { ( x + 5 ) ^ { 2 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x + 5 ) ^ { - 2 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 5 } { ( x + 5 ) ^ { 2 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x + 5 ) ^ { 2 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { ( x + 5 ) ^ { - 2 } }$ 
A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 5 } { ( x + 5 ) ^ { 2 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x + 5 ) ^ { - 2 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 5 } { ( x + 5 ) ^ { 2 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x + 5 ) ^ { 2 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { ( x + 5 ) ^ { - 2 } }$

Question 2

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$

Accepted Answer

A)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 48$     
B)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = - 64$     
C)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 64$     
D)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 4$     
E)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = - 48$    
A)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 48$     
B)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = - 64$     
C)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 64$     
D)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = 4$     
E)  $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 } = - 48$

Question 3

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { 1 - e ^ { - 8 x } } { x }$

Accepted Answer

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

Question 4

Find the limit of the sequence (if it exists).   $\alpha _ { n } = \frac { ( n + 6 ) ! } { n ! }$

Accepted Answer

A)  $\lim_ { n \rightarrow \infty }\alpha _ { n } = - 6$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
C)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 1$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 6$ 
E)  $\lim _ { n \rightarrow \infty }a _ { n } = \infty$ 
A)  $\lim_ { n \rightarrow \infty }\alpha _ { n } = - 6$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
C)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 1$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 6$ 
E)  $\lim _ { n \rightarrow \infty }a _ { n } = \infty$

Question 5

Find $\lim _ { x \rightarrow 2 } \frac { 9 x } { \sqrt { x + 9 } }$ by direct substitution.

Accepted Answer

A) 1 
B)  $18 \sqrt { 11 }$ 
C)  $\frac { 18 } { \sqrt { 11 } }$ 
D)  $\sqrt { 11 }$ 
E)  $18$ 
A) 1 
B)  $18 \sqrt { 11 }$ 
C)  $\frac { 18 } { \sqrt { 11 } }$ 
D)  $\sqrt { 11 }$ 
E)  $18$

Question 6

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

Accepted Answer

A)  $\frac { \sqrt { 34 } } { 75 } - 625$ 
B)  $\frac { \sqrt { 34 } } { 75 } + 625$ 
C)  $\frac { \sqrt { 34 } } { 75 }$ 
D) limit does not exist 
E)  $\frac { 1 } { 75 } - 625$ 
A)  $\frac { \sqrt { 34 } } { 75 } - 625$ 
B)  $\frac { \sqrt { 34 } } { 75 } + 625$ 
C)  $\frac { \sqrt { 34 } } { 75 }$ 
D) limit does not exist 
E)  $\frac { 1 } { 75 } - 625$

Question 7

Find the derivative of the function.   $f ( x ) = \frac { 1 } { x - 10 }$

Accepted Answer

A)  $- \frac { 1 } { 2 \sqrt { x + 10 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 10 } }$ 
C)  $- \frac { 1 } { ( x - 10 ) ^ { 2 } }$ 
D)  $\frac { 1 } { ( x + 10 ) ^ { 2 } }$ 
E)  $\frac { 30 } { x ^ { 4 } }$ 
A)  $- \frac { 1 } { 2 \sqrt { x + 10 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 10 } }$ 
C)  $- \frac { 1 } { ( x - 10 ) ^ { 2 } }$ 
D)  $\frac { 1 } { ( x + 10 ) ^ { 2 } }$ 
E)  $\frac { 30 } { x ^ { 4 } }$

Question 8

Find the limit (if it exists).   $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x }$

Accepted Answer

A)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = - 16$ 
B)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = - 4$ 
C)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 20$ 
D)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 16$ 
E)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 4$ 
A)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = - 16$ 
B)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = - 4$ 
C)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 20$ 
D)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 16$ 
E)  $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x } = 4$

Question 9

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \cos 2 x - 1 } { 2 x }$

Accepted Answer

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

Question 10

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 3 + 5 x } { 5 + 5 x } \right)$

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B) -1 
C)  $- \frac { 3 } { 5 }$ 
D) 1 
E) does not exist 
A)  $\frac { 3 } { 5 }$ 
B) -1 
C)  $- \frac { 3 } { 5 }$ 
D) 1 
E) does not exist

Question 11

Find the limit by direct substitution.   $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$

Accepted Answer

A)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = $\frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = $\infty$ 
C)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = - 8 
D)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = - $\frac { 1 } { 8 }$ 
E)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = 8 
A)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = $\frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = $\infty$ 
C)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = - 8 
D)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = - $\frac { 1 } { 8 }$ 
E)  $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$ = 8

Question 12

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).Round your answer to four decimal places.  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { \sqrt { 2 } } { 4 } \approx 0.3536$ 
B)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 2 \sqrt { 4 } } { 4 } \approx 1$ 
C)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = - \frac { 2 } { 4 } \approx - 0.5$ 
D)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 2 } { 4 } \approx 0.5$ 
E)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 4 \sqrt { 2 } } { 2 } \approx 2.8284$ 
A)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { \sqrt { 2 } } { 4 } \approx 0.3536$ 
B)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 2 \sqrt { 4 } } { 4 } \approx 1$ 
C)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = - \frac { 2 } { 4 } \approx - 0.5$ 
D)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 2 } { 4 } \approx 0.5$ 
E)  $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x } = \frac { 4 \sqrt { 2 } } { 2 } \approx 2.8284$

Question 13

Use asymptotes to match $f ( x ) = \frac { 3 x ^ { 2 } } { x ^ { 2 } - 2 }$ with its graph.

Accepted Answer

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

Question 14

Find the limit of the sequence. 
 $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

Accepted Answer

A)  $\frac { 4 } { 5 }$ 
B)  $\frac { 5 } { 4 }$ 
C) 3 
D) 4 
E) does not exist 
A)  $\frac { 4 } { 5 }$ 
B)  $\frac { 5 } { 4 }$ 
C) 3 
D) 4 
E) does not exist

Question 15

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.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 3 x - 4&[2,5]\
\end{array}$

Accepted Answer

A) A = 21.5 square units 
B) A = 21 square units 
C) A = 19.5 square units 
D) A = 20 square units 
E) A = 20.5 square units 
A) A = 21.5 square units 
B) A = 21 square units 
C) A = 19.5 square units 
D) A = 20 square units 
E) A = 20.5 square units

Question 16

Find the limit (if it exists).Round your answer to four decimal places.   $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z }$

Accepted Answer

A)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 0.1890$ 
B)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx - 0.3780$ 
C)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 0.3780$ 
D)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx - 0.1890$ 
E)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 5.2915$ 
A)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 0.1890$ 
B)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx - 0.3780$ 
C)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 0.3780$ 
D)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx - 0.1890$ 
E)  $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z } \approx 5.2915$

Question 17

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \sin 5 \pi x$ , $\lim _ { x \rightarrow 2 } f ( x )$

Accepted Answer

A)     $\lim _ { x \rightarrow 2 } f ( x ) = - 2$ 
B)     $\lim _ { x \rightarrow 2 } f ( x ) = 5$ 
C)    $\lim _ { x \rightarrow 2 } f ( x ) = 2$ 
D)     $\lim _ { x \rightarrow 2 } f ( x ) = 0$ 
E)     $\lim _ { x \rightarrow 2 } f ( x )$ does not exist 
A)     $\lim _ { x \rightarrow 2 } f ( x ) = - 2$ 
B)     $\lim _ { x \rightarrow 2 } f ( x ) = 5$ 
C)    $\lim _ { x \rightarrow 2 } f ( x ) = 2$ 
D)     $\lim _ { x \rightarrow 2 } f ( x ) = 0$ 
E)     $\lim _ { x \rightarrow 2 } f ( x )$ does not exist

Question 18

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 7 x \right)$ by direct substitution.

Accepted Answer

A) -30 
B) 48 
C) 12 
D) -34 
E) limit does not exist 
A) -30 
B) 48 
C) 12 
D) -34 
E) limit does not exist

Question 19

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 20

The average typing speed S (in words per minute)for a student after t weeks of lessons is given by 
 $S = \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } }$ 
What is the limit of S as t approaches infinity?

Accepted Answer

A)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = \frac { 1 } { 66 }$ 
B)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 66$ 
C)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 0$ 
D)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 160$ 
E) does not exist 
A)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = \frac { 1 } { 66 }$ 
B)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 66$ 
C)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 0$ 
D)  $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 160$ 
E) does not exist

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 1 } { x + 5 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { 1 - e ^ { - 8 x } } { x }$

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { ( n + 6 ) ! } { n ! }$

Find $\lim _ { x \rightarrow 2 } \frac { 9 x } { \sqrt { x + 9 } }$ by direct substitution.

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

Find the derivative of the function. $f ( x ) = \frac { 1 } { x - 10 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \cos 2 x - 1 } { 2 x }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 3 + 5 x } { 5 + 5 x } \right)$

Find the limit by direct substitution. $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).Round your answer to four decimal places. $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }$

Use asymptotes to match $f ( x ) = \frac { 3 x ^ { 2 } } { x ^ { 2 } - 2 }$ with its graph.

Find the limit of the sequence. $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

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)=3x-4 [2,5]

Find the limit (if it exists).Round your answer to four decimal places. $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \sin 5 \pi x$ , $\lim _ { x \rightarrow 2 } f ( x )$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 7 x \right)$ by direct substitution.

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

The average typing speed S (in words per minute)for a student after t weeks of lessons is given by $S = \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } }$ What is the limit of S as t approaches infinity?

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

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Exam 12: Limits and An Introduction To Calculus

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)​ . f(x)=1x+5f ( x ) = \frac { 1 } { x + 5 }f(x)=x+51​

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡t→4t3−64t−4\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }limt→4​t−4t3−64​

Select the correct graph for the following function using a graphing utility. f(x)=1−e−8xxf ( x ) = \frac { 1 - e ^ { - 8 x } } { x }f(x)=x1−e−8x​

Find the limit of the sequence (if it exists). αn=(n+6)!n!\alpha _ { n } = \frac { ( n + 6 ) ! } { n ! }αn​=n!(n+6)!​

Find lim⁡x→29xx+9\lim _ { x \rightarrow 2 } \frac { 9 x } { \sqrt { x + 9 } }limx→2​x+9​9x​ by direct substitution.

Find lim⁡x→5[g(x)−f(x)]\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]limx→5​[g(x)−f(x)] for f(x)=5x3f ( x ) = 5 x ^ { 3 }f(x)=5x3 and g(x)=x2+93x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 3 x ^ { 2 } }g(x)=3x2x2+9​​ .

Find the derivative of the function. f(x)=1x−10f ( x ) = \frac { 1 } { x - 10 }f(x)=x−101​

Find the limit (if it exists). lim⁡x→π/24(cos⁡x−1)sin⁡x\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x }limx→π/2​sinx4(cosx−1)​

Select the correct graph for the following function using a graphing utility. f(x)=cos⁡2x−12xf ( x ) = \frac { \cos 2 x - 1 } { 2 x }f(x)=2xcos2x−1​

Find the limit (if it exists). lim⁡x→∞(3+5x5+5x)\lim _ { x \rightarrow \infty } \left( \frac { 3 + 5 x } { 5 + 5 x } \right)limx→∞​(5+5x3+5x​)

Find the limit by direct substitution. lim⁡x→6(−48x)\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)limx→6​(−x48​)

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).Round your answer to four decimal places. lim⁡x→0−x+2−2x\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }limx→0−​xx+2​−2​​

Use asymptotes to match f(x)=3x2x2−2f ( x ) = \frac { 3 x ^ { 2 } } { x ^ { 2 } - 2 }f(x)=x2−23x2​ with its graph.

Find the limit of the sequence. 1n(n+1n[n(n+3)4])\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)n1​(n+n1​[4n(n+3)​])

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)=3x-4 [2,5]

Find the limit (if it exists).Round your answer to four decimal places. lim⁡z→07−z−7z\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z }limz→0​z7−z​−7​​

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. f(x)=sin⁡5πxf ( x ) = \sin 5 \pi xf(x)=sin5πx , lim⁡x→2f(x)\lim _ { x \rightarrow 2 } f ( x )limx→2​f(x)

Find lim⁡x→−3(13x3−7x)\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 7 x \right)limx→−3​(31​x3−7x) by direct substitution.

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+6xf ( x ) = - 4 + 6 xf(x)=−4+6x .

The average typing speed S (in words per minute)for a student after t weeks of lessons is given by S=160t266+t2S = \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } }S=66+t2160t2​ What is the limit of S as t approaches infinity?

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 $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 1 } { x + 5 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { t \rightarrow 4 } \frac { t ^ { 3 } - 64 } { t - 4 }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { 1 - e ^ { - 8 x } } { x }$

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { ( n + 6 ) ! } { n ! }$

Find $\lim _ { x \rightarrow 2 } \frac { 9 x } { \sqrt { x + 9 } }$ by direct substitution.

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

Find the derivative of the function. $f ( x ) = \frac { 1 } { x - 10 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \pi / 2 } \frac { 4 ( \cos x - 1 ) } { \sin x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \cos 2 x - 1 } { 2 x }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 3 + 5 x } { 5 + 5 x } \right)$

Find the limit by direct substitution. $\lim _ { x \rightarrow 6 } \left( - \frac { 48 } { x } \right)$

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).Round your answer to four decimal places. $\lim _ { x \rightarrow 0 ^ { - } } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }$

Use asymptotes to match $f ( x ) = \frac { 3 x ^ { 2 } } { x ^ { 2 } - 2 }$ with its graph.

Find the limit of the sequence. $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

Find the limit (if it exists).Round your answer to four decimal places. $\lim _ { z \rightarrow 0 } \frac { \sqrt { 7 - z } - \sqrt { 7 } } { z }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \sin 5 \pi x$ , $\lim _ { x \rightarrow 2 } f ( x )$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 7 x \right)$ by direct substitution.

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

The average typing speed S (in words per minute)for a student after t weeks of lessons is given by $S = \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } }$ What is the limit of S as t approaches infinity?