Question 1

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$.

Accepted Answer

A)  $- 7$ 
B)  $- 11$ 
C)  7 
D)  $- 4$ 
E)  limit does not exist 
A)  $- 7$ 
B)  $- 11$ 
C)  7 
D)  $- 4$ 
E)  limit does not exist C

Question 2

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals.
$$f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$$

Accepted Answer

A)  $5.71$ square units 
B)  $2.86$ square units 
C)  $3.69$ square units 
D)  $8.57$ square units 
E)  $17.13$ square units 
A)  $5.71$ square units 
B)  $2.86$ square units 
C)  $3.69$ square units 
D)  $8.57$ square units 
E)  $17.13$ square units B

Question 3

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$$

Accepted Answer

A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist. A

Question 4

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$, if it exists.

Accepted Answer

A)  $- 1$ 
B)  $- 2$ 
C)  2 
D)  1 
E)  The limit does not exist. 
A)  $- 1$ 
B)  $- 2$ 
C)  2 
D)  1 
E)  The limit does not exist.

Question 5

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$.

Accepted Answer

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

Question 6

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0 
A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0

Question 7

Find
$$\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$$
by direct substitution.

Accepted Answer

A)  $- 15$ 
B)  33 
C)  $- 3$ 
D)  $- 29$ 
E)  limit does not exist 
A)  $- 15$ 
B)  33 
C)  $- 3$ 
D)  $- 29$ 
E)  limit does not exist

Question 8

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Accepted Answer

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

Question 9

A union contract guarantees a $16 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,500$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is given by
$$f ( t ) = \left\{ \begin{array} { l l } 
31.50 , & 0 < t \leq 1 \
36.54 , & 1 < t \leq 2 \
42.39 , & 2 < t \leq 3
\end{array} \right.$$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.

Accepted Answer

A)  $31.50$ 
B)  $36.54$ 
C)  $1.00$ 
D)  $42.39$ 
E)  The limit does not exist. 
A)  $31.50$ 
B)  $36.54$ 
C)  $1.00$ 
D)  $42.39$ 
E)  The limit does not exist.

Question 10

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }$ (assume $n$ begins with 1).

Accepted Answer

A)  $\frac { 8 } { 3 }$ 
B)  183 
C)  126 
D)  0 
E)  the sequence diverges 
A)  $\frac { 8 } { 3 }$ 
B)  183 
C)  126 
D)  0 
E)  the sequence diverges

Question 11

Approximate the area of the indicated region under the given curve using five rectangles.
$$f ( x ) = 5 - x ^ { 2 }$$

Accepted Answer

A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $9.25$ 
A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $9.25$

Question 12

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$.

Accepted Answer

A)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 9 } { 4 }$ 
B)  $S ( n ) = \frac { 9 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 9$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 36 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$, the limit does not exist 
A)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 9 } { 4 }$ 
B)  $S ( n ) = \frac { 9 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 9$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 36 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$, the limit does not exist

Question 13

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 14

Find the following limit, if it exists.
$\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Accepted Answer

A)  $- 3$ 
B)  $- 15$ 
C)  3 
D)  $- 243$ 
E)  The limit does not exist. 
A)  $- 3$ 
B)  $- 15$ 
C)  3 
D)  $- 243$ 
E)  The limit does not exist.

Question 15

Find the following limit. Round your answer to two decimals.
$\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

Accepted Answer

A)  $0.22$ 
B)  $1.32$ 
C)  $0.44$ 
D)  $0.18$ 
E)  $1.41$ 
A)  $0.22$ 
B)  $1.32$ 
C)  $0.44$ 
D)  $0.18$ 
E)  $1.41$

Question 16

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

Accepted Answer

A)  3 
B)  $\frac { 1 } { 2 }$ 
C)  $- 3$ 
D)  0 
E)  The slope is undefined. 
A)  3 
B)  $\frac { 1 } { 2 }$ 
C)  $- 3$ 
D)  0 
E)  The slope is undefined.

Question 17

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 18

A union contract guarantees a $15 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,000$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is given by
$$f ( t ) = \left\{ \begin{array} { l l } 
31.00 , & 0 < t \leq 1 \
35.65 , & 1 < t \leq 2 \
41.00 , & 2 < t \leq 3
\end{array} \right.$$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.

Accepted Answer

A)  $31.00$ 
B)  $35.65$ 
C)  $1.00$ 
D)  $41.00$ 
E)  The limit does not exist. 
A)  $31.00$ 
B)  $35.65$ 
C)  $1.00$ 
D)  $41.00$ 
E)  The limit does not exist.

Question 19

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

Accepted Answer

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

Question 20

Find the following limit, if it exists.
$$\lim _ { x \rightarrow 3 } x ^ { 7 }$$

Accepted Answer

A)  3 
B)  21 
C)  $- 3$ 
D)  2187 
E)  The limit does not exist. 
A)  3 
B)  21 
C)  $- 3$ 
D)  2187 
E)  The limit does not exist.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$ .

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals. $f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$ . $Use the graph to find \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 } .$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$ by direct substitution.

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }$ (assume $n$ begins with 1).

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

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

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

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

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

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 following limit, if it exists. $\lim _ { x \rightarrow 3 } x ^ { 7 }$

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

Exam 11: 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)​ for f(x)=−4+7xf ( x ) = - 4 + 7 xf(x)=−4+7x .

Approximate the area of the region under the function below on the interval [0,2][ 0,2 ][0,2] using 8 rectangles. Round your answer to two decimals. f(x)=13x4f ( x ) = \frac { 1 } { 3 } x ^ { 4 }f(x)=31​x4

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+6x−7\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }limx→1​x2+6x−7x−1​

Use the graph below to find lim⁡x→−2∣x+2∣x+2\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }limx→−2​x+2∣x+2∣​ , if it exists.

Use the graph to find lim⁡x→23x2−12x−2\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }limx→2​x−23x2−12​ .

Find the following limit, if it exists. lim⁡x→∞3x−6\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }limx→∞​x−63​

Find lim⁡x→−3(13x3−2x)\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)limx→−3​(31​x3−2x) by direct substitution.

Find lim⁡x→0−xx+7−7\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }limx→0−​x+7​−7​x​

Use the first six terms to predict the limit of the sequence an=8n3+6n+3a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }an​=n+38n3+6​ (assume nnn begins with 1).

Approximate the area of the indicated region under the given curve using five rectangles. f(x)=5−x2f ( x ) = 5 - x ^ { 2 }f(x)=5−x2

Rewrite ∑i=1n9i3n5\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }∑i=1n​n59i3​ as a rational function S(n)S ( n )S(n) and find lim⁡n→∞S(n)\lim _ { n \rightarrow \infty } S ( n )limn→∞​S(n) .

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.

Find the following limit, if it exists. lim⁡x→−3x5\lim _ { x \rightarrow - 3 } x ^ { 5 }limx→−3​x5

Find the following limit. Round your answer to two decimals. lim⁡x→3x2+72x2\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }limx→3​2x2x2+7​​

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

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

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 following limit, if it exists. lim⁡x→3x7\lim _ { x \rightarrow 3 } x ^ { 7 }limx→3​x7

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 $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$ .

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals. $f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$ . $Use the graph to find \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 } .$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$ by direct substitution.

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }$ (assume $n$ begins with 1).

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

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

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

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

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

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 following limit, if it exists. $\lim _ { x \rightarrow 3 } x ^ { 7 }$