Question 1

The limit of a function as x approaches c does not exist if the function approaches $- 5$ from the left of c and 4 from the right of

Accepted Answer

A) True 
 B)False

Question 2

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { 1 + 3 x }$ (if it exists).

Accepted Answer

A) 0 
B)   $- \frac { 2 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D) -$\infty$ 
E) does not exist

Question 3

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 32 x - 1 } { 4 x + 4 } \right)$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 32 }$ 
C) 8 
D) -8 
E) does not exist

Question 4

Find the derivative of the function.  $h ( s ) = \frac { 1 } { \sqrt { s + 2 } }$

Accepted Answer

A)  $\frac { 1 } { 2 ( s - 2 ) ^ { \frac { 3 } { 2 } } }$ 
B)  $\frac { 1 } { 2 ( s + 2 ) ^ { \frac { 3 } { 2 } } }$ 
C)  $\frac { 1 } { ( s + 2 ) ^ { 2 } }$ 
D)  $\frac { 1 } { 2 \sqrt { s - 2 } }$ 
E)  $- \frac { 1 } { 2 ( s + 2 ) ^ { \frac { 3 } { 2 } } }$

Question 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.   $h ( x ) = \sqrt { x },(4,2)$

Accepted Answer

A)  $m = \frac { 1 } { 4 }$ 
B)  $m = - 2$ 
C)  $m = 4$ 
D)  $m = - \frac { 1 } { 4 }$ 
E)  $m = 2$

Question 6

Select the correct function for the graph using horizontal asymptotes as aids.

Accepted Answer

A)  $\frac { 6 x ^ { 2 } } { x ^ { 2 } + 8 }$ 
B)  $\frac { x ^ { 2 } } { x ^ { 2 } - 6 }$ 
C)  $\frac { x ^ { 2 } } { x ^ { 2 } + 6 }$ 
D)  $\frac { 8 x ^ { 2 } } { x ^ { 2 } + 6 }$ 
E)  $\frac { 8 x ^ { 2 } } { x ^ { 2 } - 6 }$

Question 7

Use the graph to determine the limit visually (if it exists).Then identify another function f2(x)that agrees with the given function at all but one point.
  $$f ( x ) = \frac { x ^ { 2 } - 1 } { x + 1 }$$      $\lim _ { x \rightarrow 2 } f ( x ) = ?$

Accepted Answer

A)  $f _ { 2 } ( x ) = x - 1$ $\lim _ { x \rightarrow 2 } f ( x ) = 1$ 
B)  $f _ { 2 } ( x ) = 2 x - 1$ $\lim _ { x \rightarrow 2 } f ( x ) = 3$ 
C)  $f _ { 2 } ( x ) = 2 x + 1$ $\lim _ { x \rightarrow 2 } f ( x ) = 5$ 
D)  $f _ { 2 } ( x ) = x + 1$ $\lim _ { x \rightarrow 2 } f ( x ) = 5$ 
E)  $f _ { 2 } ( x ) = x + 1$ $\lim _ { x \rightarrow 2 } f ( x ) = 3$

Question 8

Find the derivative of the function.   $f ( x ) = \frac { 1 } { x + 13 }$

Accepted Answer

A)  $- \frac { 1 } { ( x + 13 ) ^ { 2 } }$ 
B)  $- \frac { 1 } { 2 \sqrt { x + 13 } }$ 
C)  $\frac { 39 } { x ^ { 4 } }$ 
D)  $\frac { 1 } { ( x + 13 ) ^ { 2 } }$ 
E)  $\frac { 1 } { 2 \sqrt { x - 13 } }$

Question 9

Select the correct function for the graph using horizontal asymptotes as aids.

Accepted Answer

A)  $\frac { 6 x ^ { 2 } } { x ^ { 2 } + 5 }$ 
B)  $\frac { 5 x ^ { 2 } } { x ^ { 2 } - 6 }$ 
C)  $\frac { x ^ { 2 } } { x ^ { 2 } - 6 }$ 
D)  $\frac { x ^ { 2 } } { x ^ { 2 } + 6 }$ 
E)  $\frac { 5 x ^ { 2 } } { x ^ { 2 } + 6 }$

Question 10

Determine $\lim _ { x \rightarrow 1 } f ( x )$ where $f ( x ) = \left\{ \begin{array} { l } 
10 - x ^ { 2 } , x \leq 1 \
7 - x , x > 1
\end{array} \right.$ (if it exists)by evaluating the corresponding one-sided limits.

Accepted Answer

A) 9 
B) 6 
C) 10 
D) 7 
E) limit does not exist

Question 11

Evaluate the sum using the summation formula and property. 
 $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

Accepted Answer

A) 9,355 
B) 9,455 
C) 9,655 
D) 9,255 
E) 9,555

Question 12

Find $\lim _ { x \rightarrow + \infty } \left[ \frac { x } { ( 3 + x ) ^ { 2 } } - 4 \right]$ (if it exists).

Accepted Answer

A)   $- \frac { 1 } { 3 }$ 
B) 4 
C) -4 
D)  $\infty$ 
E) does not exist

Question 13

Select the correct graph of the following function.  $f ( x ) = 5 x - \sqrt { 25 x ^ { 2 } + 3 }$

Accepted Answer

A)   
B)   
C)   
D)    
E)

Question 14

Find the limit by direct substitution.Round your answer to two decimal places.   $\lim _ { x \rightarrow 1 } e ^ { 3 x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 1 } e ^ { 3 x }$ = 8103.08 
B)  $\lim _ { x \rightarrow 1 } e ^ { 3 x }$ = 2.72 
C)  $\lim _ { x \rightarrow 1 } e ^ { 3 x }$ = $\infty$ 
D)  $\lim _ { x \rightarrow 1 } e ^ { 3 x }$ = 20.09 
E)  $\lim _ { x \rightarrow 1 } e ^ { 3 x }$ = -20.09

Question 15

Evaluate the sum using the summation formula and property. 
 $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

Accepted Answer

A) 3,015 
B) 3,315 
C) 3,215 
D) 2,915 
E) 3,115

Question 16

Find the limit 
 $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } = 5$ 
B)  $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } = 4$ 
C)  $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } = - 3$ 
D)  $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } = - 5$ 
E)  $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } = 3$

Question 17

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.   $h ( x ) = 5 x + 7 , \quad ( - 1,2 )$

Accepted Answer

A)  $m = - 7$ 
B)  $m = 7$ 
C)  $m = 4$ 
D)  $m = 5$ 
E)  $m = - 2$

Question 18

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

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 5 } { 2 \sqrt { x } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 7 } { 2 \sqrt { x } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 3 } { 2 \sqrt { x } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 5 } { 2 \sqrt { x } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 7 } { 2 \sqrt { x } }$

Question 19

Find the limit by direct substitution.   $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$

Accepted Answer

A)  $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$ = $\frac { 1 } { 19 }$ 
B)  $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$ = -19 
C)  $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$ = $\infty$ 
D)  $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$ = $\frac { 19 } { 2 }$ 
E)  $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$ = $- \frac { 2 } { 19 }$

Question 20

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.   $g ( x ) = 10 - 2 x , \quad ( 1,8 )$

Accepted Answer

A)  $m = 8$ 
B)  $m = - 2$ 
C)  $m = 10$ 
D)  $m = 7$ 
E)  $m = - 10$

Exam 12: Limits and An Introduction To Calculus

The limit of a function as x approaches c does not exist if the function approaches $- 5$ from the left of c and 4 from the right of

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { 1 + 3 x }$ (if it exists) .

Find the limit (if it exists) . $\lim _ { x \rightarrow \infty } \left( \frac { 32 x - 1 } { 4 x + 4 } \right)$

Find the derivative of the function. $h ( s ) = \frac { 1 } { \sqrt { s + 2 } }$

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. $h ( x ) = \sqrt { x },(4,2)$

Find the derivative of the function. $f ( x ) = \frac { 1 } { x + 13 }$

Determine $\lim _ { x \rightarrow 1 } f ( x )$ where $f ( x ) = \left\{ \begin{array} { l } 10 - x ^ { 2 } , x \leq 1 \\7 - x , x > 1\end{array} \right.$ (if it exists) by evaluating the corresponding one-sided limits.

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

Find $\lim _ { x \rightarrow + \infty } \left[ \frac { x } { ( 3 + x ) ^ { 2 } } - 4 \right]$ (if it exists) .

Select the correct graph of the following function. $f ( x ) = 5 x - \sqrt { 25 x ^ { 2 } + 3 }$

Find the limit by direct substitution.Round your answer to two decimal places. $\lim _ { x \rightarrow 1 } e ^ { 3 x }$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

Find the limit $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x }$

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. $h ( x ) = 5 x + 7 , \quad ( - 1,2 )$

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

Find the limit by direct substitution. $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$

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. $g ( x ) = 10 - 2 x , \quad ( 1,8 )$

Exam 12: Limits and An Introduction To Calculus

The limit of a function as x approaches c does not exist if the function approaches −5- 5−5 from the left of c and 4 from the right of

Find lim⁡x→+∞1−2x1+3x\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { 1 + 3 x }limx→+∞​1+3x1−2x​ (if it exists) .

Find the limit (if it exists) . lim⁡x→∞(32x−14x+4) \lim _ { x \rightarrow \infty } \left( \frac { 32 x - 1 } { 4 x + 4 } \right) limx→∞​(4x+432x−1​)

Find the derivative of the function. h(s) =1s+2h ( s ) = \frac { 1 } { \sqrt { s + 2 } }h(s) =s+2​1​

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. h(x) =x,(4,2) h ( x ) = \sqrt { x },(4,2) h(x) =x​,(4,2)

Select the correct function for the graph using horizontal asymptotes as aids.

Find the derivative of the function. f(x) =1x+13f ( x ) = \frac { 1 } { x + 13 }f(x) =x+131​

Select the correct function for the graph using horizontal asymptotes as aids.

Evaluate the sum using the summation formula and property. ∑i=130i2\sum _ { i = 1 } ^ { 30 } i ^ { 2 }∑i=130​i2

Find lim⁡x→+∞[x(3+x) 2−4]\lim _ { x \rightarrow + \infty } \left[ \frac { x } { ( 3 + x ) ^ { 2 } } - 4 \right]limx→+∞​[(3+x) 2x​−4] (if it exists) .

Select the correct graph of the following function. f(x) =5x−25x2+3f ( x ) = 5 x - \sqrt { 25 x ^ { 2 } + 3 }f(x) =5x−25x2+3​

Find the limit by direct substitution.Round your answer to two decimal places. lim⁡x→1e3x\lim _ { x \rightarrow 1 } e ^ { 3 x }limx→1​e3x

Evaluate the sum using the summation formula and property. ∑k=110(k3+9) \sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right) ∑k=110​(k3+9)

Find the limit lim⁡x→0sin⁡3xx\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x }limx→0​xsin3x​

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. h(x) =5x+7,(−1,2) h ( x ) = 5 x + 7 , \quad ( - 1,2 ) h(x) =5x+7,(−1,2)

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) =5xf ( x ) = 5 \sqrt { x }f(x) =5x​

Find the limit by direct substitution. lim⁡x→4(x−1x2+8x+9) \lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right) limx→4​(x2+8x+9x−1​)

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) =10−2x,(1,8) g ( x ) = 10 - 2 x , \quad ( 1,8 ) g(x) =10−2x,(1,8)

The limit of a function as x approaches c does not exist if the function approaches $- 5$ from the left of c and 4 from the right of

Find $\lim _ { x \rightarrow + \infty } \frac { 1 - 2 x } { 1 + 3 x }$ (if it exists) .

Find the limit (if it exists) . $\lim _ { x \rightarrow \infty } \left( \frac { 32 x - 1 } { 4 x + 4 } \right)$

Find the derivative of the function. $h ( s ) = \frac { 1 } { \sqrt { s + 2 } }$

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. $h ( x ) = \sqrt { x },(4,2)$

Find the derivative of the function. $f ( x ) = \frac { 1 } { x + 13 }$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

Find $\lim _ { x \rightarrow + \infty } \left[ \frac { x } { ( 3 + x ) ^ { 2 } } - 4 \right]$ (if it exists) .

Select the correct graph of the following function. $f ( x ) = 5 x - \sqrt { 25 x ^ { 2 } + 3 }$

Find the limit by direct substitution.Round your answer to two decimal places. $\lim _ { x \rightarrow 1 } e ^ { 3 x }$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

Find the limit $\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x }$

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. $h ( x ) = 5 x + 7 , \quad ( - 1,2 )$

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

Find the limit by direct substitution. $\lim _ { x \rightarrow 4 } \left( \frac { x - 1 } { x ^ { 2 } + 8 x + 9 } \right)$

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. $g ( x ) = 10 - 2 x , \quad ( 1,8 )$