Question 1

Find the limit by direct substitution.   $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = - \frac { 5 } { 4 }$ 
B)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = \frac { 5 } { 4 }$ 
C)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }$ = $\infty$ 
D)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = \frac { 4 } { 5 }$ 
E)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = - \frac { 4 } { 5 }$ 
A)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = - \frac { 5 } { 4 }$ 
B)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = \frac { 5 } { 4 }$ 
C)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }$ = $\infty$ 
D)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = \frac { 4 } { 5 }$ 
E)  $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 } = - \frac { 4 } { 5 }$

Question 2

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point.   $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Accepted Answer

A)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
D)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
A)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
D)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$

Question 3

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 3 }$ .

Accepted Answer

A)  $\frac { 1 } { 3 \sqrt { x - 3 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 3 } }$ 
C)  $\frac { \sqrt { x - 3 } } { 2 }$ 
D)  $\sqrt { x - 3 }$ 
E) limit does not exist 
A)  $\frac { 1 } { 3 \sqrt { x - 3 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 3 } }$ 
C)  $\frac { \sqrt { x - 3 } } { 2 }$ 
D)  $\sqrt { x - 3 }$ 
E) limit does not exist

Question 4

Evaluate $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Accepted Answer

A) 44,100 
B) 42,500 
C) 44,096 
D) 44,020 
E) 32,620 
A) 44,100 
B) 42,500 
C) 44,096 
D) 44,020 
E) 32,620

Question 5

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 4 } { x } - 6 \right)$

Accepted Answer

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

Question 6

Select the correct graph for the following function and find the limit (if it exists)as x approaches 2.  $f ( x ) = \left\{ \begin{array} { l } 
5 x + 1 , x < 2 \
x + 9 , x \geq 2
\end{array} \right.$

Accepted Answer

A)     The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = 9$ 
B)       $\lim _ { x \rightarrow2 } f ( x ) \text { does not exist }$ . 
C)      $\lim _ { x \rightarrow2 } f ( x ) \text { does not exist }$ 
D)     The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = 11$ 
E)      The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = - 11$ 
A)     The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = 9$ 
B)       $\lim _ { x \rightarrow2 } f ( x ) \text { does not exist }$ . 
C)      $\lim _ { x \rightarrow2 } f ( x ) \text { does not exist }$ 
D)     The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = 11$ 
E)      The limit exists as x approaches 2: $\lim _ { x \rightarrow 2 } f ( x ) = - 11$

Question 7

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \frac { 1 } { 5 }$     
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = 5$     
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = \frac { 1 } { 5 }$    
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \frac { 1 } { \sqrt { 5 } }$     
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \sqrt { 5 }$    
A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \frac { 1 } { 5 }$     
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = 5$     
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = \frac { 1 } { 5 }$    
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \frac { 1 } { \sqrt { 5 } }$     
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x } = - \sqrt { 5 }$

Question 8

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { e ^ { 4 x } - 1 } { 4 x }$

Accepted Answer

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

Question 9

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

Accepted Answer

A) 0 
B) limit does not exist 
C) -1 
D) 2 
E) 4 
A) 0 
B) limit does not exist 
C) -1 
D) 2 
E) 4

Question 10

Find the derivative of the function.   $f ( x ) = 8$

Accepted Answer

A)  $8 x$ 
B)  $- 8 x$ 
C)  $0$ 
D)  $- 8$ 
E)  $8$ 
A)  $8 x$ 
B)  $- 8 x$ 
C)  $0$ 
D)  $- 8$ 
E)  $8$

Question 11

Find the derivative of $f ( x ) = 2 x ^ { 2 } + 4 x + 9$ .

Accepted Answer

A)  $4 x + 13$ 
B)  $2 x + 13$ 
C)  $2 x + 4$ 
D)  $4 x + 4$ 
E)  $4 x$ 
A)  $4 x + 13$ 
B)  $2 x + 13$ 
C)  $2 x + 4$ 
D)  $4 x + 4$ 
E)  $4 x$

Question 12

Find the limit (if it exists).  $\lim _ { t \rightarrow + \infty } \frac { t ^ { 2 } } { t + 2 }$

Accepted Answer

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

Question 13

Find a formula for the slope of the graph of $f ( x ) = \frac { 7 } { x + 4 }$ .

Accepted Answer

A)  $\frac { 7 } { ( x + 4 ) ^ { 2 } }$ 
B)  $\frac { 7 } { x + 4 }$ 
C)  $- \frac { 1 } { 7 ( x + 4 ) ^ { 2 } }$ 
D)  $- \frac { 7 } { ( x + 4 ) ^ { 2 } }$ 
E)  $- \frac { 7 } { x + 4 }$ 
A)  $\frac { 7 } { ( x + 4 ) ^ { 2 } }$ 
B)  $\frac { 7 } { x + 4 }$ 
C)  $- \frac { 1 } { 7 ( x + 4 ) ^ { 2 } }$ 
D)  $- \frac { 7 } { ( x + 4 ) ^ { 2 } }$ 
E)  $- \frac { 7 } { x + 4 }$

Question 14

Find $\lim _{y \rightarrow 0} \frac{\sqrt{10+y}-\sqrt{10}}{y}$ .

Accepted Answer

A)  $\frac { \sqrt { 10 } } { 20 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C) 0 
D)  $\frac { \sqrt { 10 } } { 2 }$ 
E) limit does not exist 
A)  $\frac { \sqrt { 10 } } { 20 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C) 0 
D)  $\frac { \sqrt { 10 } } { 2 }$ 
E) limit does not exist

Question 15

Find $\lim _ { x \rightarrow 4 } f ( x )$ .   $f ( x ) = \frac { x } { 5 - x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 4 } f ( x )$ = -2 
B)  $\lim _ { x \rightarrow 4 } f ( x )$ = $\infty$ 
C)  $\lim _ { x \rightarrow 4 } f ( x )$ = 4 
D)  $\lim _ { x \rightarrow 4 } f ( x )$ = 5 
E)  $\lim _ { x \rightarrow 4 } f ( x )$ = 2 
A)  $\lim _ { x \rightarrow 4 } f ( x )$ = -2 
B)  $\lim _ { x \rightarrow 4 } f ( x )$ = $\infty$ 
C)  $\lim _ { x \rightarrow 4 } f ( x )$ = 4 
D)  $\lim _ { x \rightarrow 4 } f ( x )$ = 5 
E)  $\lim _ { x \rightarrow 4 } f ( x )$ = 2

Question 16

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point.   $f ( x ) = \frac { 1 } { x + 4 } , \quad ( - 3,1 )$

Accepted Answer

A)  $\begin{array} { l } 
m = \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = 1
\end{array}$ 
B)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = 1
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$ 
A)  $\begin{array} { l } 
m = \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = 1
\end{array}$ 
B)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = 1
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { ( x + 4 ) ^ { 2 } } \
\text { At } ( - 3,1 ) , m = - 1
\end{array}$

Question 17

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point.   $f ( x ) = 3 - x ^ { 2 } , \quad ( - 1,2 )$

Accepted Answer

A)  $\begin{array} { l } 
m = - 2 x \
\text { At } ( - 1,2 ) , m = \frac { 1 } { 2 }
\end{array}$ 
B)  $\begin{array} { l } 
m = - 2 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
C)  $\begin{array} { l } 
m = 2 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
D)  $\begin{array} { l } 
m = - 3 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
E)  $\begin{array} { l } 
m = 2 x \
\text { At } ( - 1,2 ) , m = - 3
\end{array}$ 
A)  $\begin{array} { l } 
m = - 2 x \
\text { At } ( - 1,2 ) , m = \frac { 1 } { 2 }
\end{array}$ 
B)  $\begin{array} { l } 
m = - 2 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
C)  $\begin{array} { l } 
m = 2 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
D)  $\begin{array} { l } 
m = - 3 x \
\text { At } ( - 1,2 ) , m = 2
\end{array}$ 
E)  $\begin{array} { l } 
m = 2 x \
\text { At } ( - 1,2 ) , m = - 3
\end{array}$

Question 18

Find the limit (if it exists).  $\lim _ { x \rightarrow + \infty } \frac { 6 x ^ { 2 } - 6 } { ( x - 5 ) ^ { 2 } }$

Accepted Answer

A) 6 
B)  $\frac { 1 } { 6 }$ 
C)  $- \frac { 1 } { 6 }$ 
D) -6 
E) does not exist 
A) 6 
B)  $\frac { 1 } { 6 }$ 
C)  $- \frac { 1 } { 6 }$ 
D) -6 
E) does not exist

Question 19

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 2$ and the x-axis on the interval [0,9].

Accepted Answer

A) 747 
B) 243 
C) 731 
D) 261 
E) 245 
A) 747 
B) 243 
C) 731 
D) 261 
E) 245

Question 20

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 4 \right)$ using the summation formulas and properties.

Accepted Answer

A) 8,004 
B) 176,400 
C) 44,180 
D) 176,480 
E) 44,100 
A) 8,004 
B) 176,400 
C) 44,180 
D) 176,480 
E) 44,100

Find the limit by direct substitution. $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 3 }$ .

Evaluate $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 } { x } - 6 \right)$

Select the correct graph for the following function and find the limit (if it exists)as x approaches 2. $f ( x ) = \left\{ \begin{array} { l } 5 x + 1 , x < 2 \\x + 9 , x \geq 2\end{array} \right.$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { e ^ { 4 x } - 1 } { 4 x }$

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 the derivative of the function. $f ( x ) = 8$

Find the derivative of $f ( x ) = 2 x ^ { 2 } + 4 x + 9$ .

Find the limit (if it exists). $\lim _ { t \rightarrow + \infty } \frac { t ^ { 2 } } { t + 2 }$

Find a formula for the slope of the graph of $f ( x ) = \frac { 7 } { x + 4 }$ .

Find $\lim _{y \rightarrow 0} \frac{\sqrt{10+y}-\sqrt{10}}{y}$ .

Find $\lim _ { x \rightarrow 4 } f ( x )$ . $f ( x ) = \frac { x } { 5 - x }$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \frac { 1 } { x + 4 } , \quad ( - 3,1 )$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = 3 - x ^ { 2 } , \quad ( - 1,2 )$

Find the limit (if it exists). $\lim _ { x \rightarrow + \infty } \frac { 6 x ^ { 2 } - 6 } { ( x - 5 ) ^ { 2 } }$

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 2$ and the x-axis on the interval [0,9].

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 4 \right)$ using the summation formulas and properties.

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 limit by direct substitution. lim⁡x→15x+1x−10\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }limx→15​x−10x+1​​

Find a formula for the slope of the graph of fff at the point (x,f(x))( x , f ( x ) )(x,f(x)) .Then use it to find the slope at the given point. f(x)=x−1,(17,4)f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )f(x)=x−1​,(17,4)

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)=x−3f ( x ) = \sqrt { x - 3 }f(x)=x−3​ .

Evaluate ∑j=120(j3−4j2)\sum _ { j = 1 } ^ { 20 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)∑j=120​(j3−4j2) using the summation formulas and properties.

Find the limit (if it exists). lim⁡x→∞(4x−6)\lim _ { x \rightarrow \infty } \left( \frac { 4 } { x } - 6 \right)limx→∞​(x4​−6)

Select the correct graph for the following function and find the limit (if it exists)as x approaches 2. f(x)={5x+1,x<2x+9,x≥2f ( x ) = \left\{ \begin{array} { l } 5 x + 1 , x < 2 \\x + 9 , x \geq 2\end{array} \right.f(x)={5x+1,x<2x+9,x≥2​

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→05x+5−1x\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x }limx→0​xx+55​−1​

Select the correct graph for the following function using a graphing utility. f(x)=e4x−14xf ( x ) = \frac { e ^ { 4 x } - 1 } { 4 x }f(x)=4xe4x−1​

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 the derivative of the function. f(x)=8f ( x ) = 8f(x)=8

Find the derivative of f(x)=2x2+4x+9f ( x ) = 2 x ^ { 2 } + 4 x + 9f(x)=2x2+4x+9 .

Find the limit (if it exists). lim⁡t→+∞t2t+2\lim _ { t \rightarrow + \infty } \frac { t ^ { 2 } } { t + 2 }limt→+∞​t+2t2​

Find a formula for the slope of the graph of f(x)=7x+4f ( x ) = \frac { 7 } { x + 4 }f(x)=x+47​ .

Find lim⁡y→010+y−10y\lim _{y \rightarrow 0} \frac{\sqrt{10+y}-\sqrt{10}}{y}limy→0​y10+y​−10​​ .

Find lim⁡x→4f(x)\lim _ { x \rightarrow 4 } f ( x )limx→4​f(x) . f(x)=x5−xf ( x ) = \frac { x } { 5 - x }f(x)=5−xx​

Find a formula for the slope of the graph of fff at the point (x,f(x))( x , f ( x ) )(x,f(x)) .Then use it to find the slope at the given point. f(x)=1x+4,(−3,1)f ( x ) = \frac { 1 } { x + 4 } , \quad ( - 3,1 )f(x)=x+41​,(−3,1)

Find a formula for the slope of the graph of fff at the point (x,f(x))( x , f ( x ) )(x,f(x)) .Then use it to find the slope at the given point. f(x)=3−x2,(−1,2)f ( x ) = 3 - x ^ { 2 } , \quad ( - 1,2 )f(x)=3−x2,(−1,2)

Find the limit (if it exists). lim⁡x→+∞6x2−6(x−5)2\lim _ { x \rightarrow + \infty } \frac { 6 x ^ { 2 } - 6 } { ( x - 5 ) ^ { 2 } }limx→+∞​(x−5)26x2−6​

Use the limit process to find the area of the region between f(x)=x2+2f ( x ) = x ^ { 2 } + 2f(x)=x2+2 and the x-axis on the interval [0,9].

Evaluate ∑k=120(k3+4)\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 4 \right)∑k=120​(k3+4) using the summation formulas and properties.

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 limit by direct substitution. $\lim _ { x \rightarrow 15 } \frac { \sqrt { x + 1 } } { x - 10 }$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 3 }$ .

Evaluate $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 } { x } - 6 \right)$

Select the correct graph for the following function and find the limit (if it exists)as x approaches 2. $f ( x ) = \left\{ \begin{array} { l } 5 x + 1 , x < 2 \\x + 9 , x \geq 2\end{array} \right.$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 5 } { x + 5 } - 1 } { x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { e ^ { 4 x } - 1 } { 4 x }$

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 the derivative of the function. $f ( x ) = 8$

Find the derivative of $f ( x ) = 2 x ^ { 2 } + 4 x + 9$ .

Find the limit (if it exists). $\lim _ { t \rightarrow + \infty } \frac { t ^ { 2 } } { t + 2 }$

Find a formula for the slope of the graph of $f ( x ) = \frac { 7 } { x + 4 }$ .

Find $\lim _{y \rightarrow 0} \frac{\sqrt{10+y}-\sqrt{10}}{y}$ .

Find $\lim _ { x \rightarrow 4 } f ( x )$ . $f ( x ) = \frac { x } { 5 - x }$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \frac { 1 } { x + 4 } , \quad ( - 3,1 )$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = 3 - x ^ { 2 } , \quad ( - 1,2 )$

Find the limit (if it exists). $\lim _ { x \rightarrow + \infty } \frac { 6 x ^ { 2 } - 6 } { ( x - 5 ) ^ { 2 } }$

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 2$ and the x-axis on the interval [0,9].

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 4 \right)$ using the summation formulas and properties.