Question 1

Determine whether the sequence $a _ { \mathrm { n } } = \frac { n ^ { 4 } - 1 } { 5 - n ^ { 4 } }$ converges or diverges. If it converges, find the limit.

Accepted Answer

Converges; $\lim _ { n \rightarrow \infty } \frac { n ^ { 4 } - 1 } { 5 - n ^ { 4 } } = - 1$

Question 2

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 2 x$ over the interval $1 \leq x \leq 3$ .

Accepted Answer

Width: $\Delta x = \frac { 3 - 1 } { n } = \frac { 2 } { n }$ , REP: $x _ { k } = ( 1 ) + k \left( \frac { 2 } { n } \right) = 1 + \frac { 2 k } { n }$ , Height: $f \left( x _ { k } \right) = 2 \left( 1 + \frac { 2 k } { n } \right) = 2 + \frac { 4 k } { n }$ . $$\begin{aligned}
A & = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } f \left( x _ { k } \right) \Delta x = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left[ \left( 2 + \frac { 4 k } { n } \right) \cdot \frac { 2 } { n } \right] = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 4 } { n } + \frac { 8 k } { n ^ { 2 } } \right) \
& = \lim _ { n \rightarrow \infty } \frac { 4 } { n } \sum _ { k = 1 } ^ { n } 1 + \lim _ { n \rightarrow \infty } \frac { 8 } { n ^ { 2 } } \sum _ { k = 1 } ^ { n } k = \lim _ { n \rightarrow \infty } \left[ \frac { 4 } { n } \cdot n \right] + \lim _ { n \rightarrow \infty } \left[ \frac { 8 } { n ^ { 2 } } \cdot \frac { n ( n + 1 ) } { 2 } \right] = \lim _ { n \rightarrow \infty } [ 4 ] + \lim _ { n \rightarrow \infty } \left[ 4 + \frac { 4 } { n } \right] = 8
\end{aligned}$$

Question 3

Evaluate $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 4 x } { x - 4 }$ .

Accepted Answer

$\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 4 x } { x - 4 } = 4$

Question 4

Evaluate $\lim _ { t \rightarrow 3 } \frac { t ^ { 2 } - 9 } { \frac { 1 } { 3 } - \frac { 1 } { t } }$ .

Accepted Answer

The answer of Evaluate \(\lim _ { t \rightarrow 3...

Question 5

For the function g whose graph is given, state the value of the given quantity if it exists.   
a) $\lim _ { x \rightarrow 4 ^ { - } } g ( t )$ 
b) $\lim _ { x \rightarrow 4 ^ { + } } g ( t )$ 
c) $\lim _ { x \rightarrow 4 } g ( t )$

Accepted Answer

The answer of For the function g whose graph is...

Question 6

Find the derivative of $f ( x ) = \frac { 2 x } { x - 2 }$ at the point $\left( - 3 , \frac { \mathfrak { 6 } } {5 } \right)$ .

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 7

Find the derivative of $f ( x ) = 5 x - x ^ { 2 }$ at the point $\left( \frac { 5 } { 2 } , \frac { 25 } { 4 } \right)$ .

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 8

Evaluate $\lim _ { x \rightarrow 3/2 } \frac { 4 x ^ { 2 } - 16 x + 15 } { 2 x - 3 }$ .

Accepted Answer

The answer of Evaluate \(\lim _ { x \rightarrow 3/2...

Question 9

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 5$ over the interval $0 \leq x \leq 5$ .

Accepted Answer

Width: @#LAT-DLM& ,

Question 10

Use a graphing device to determine whether $\lim _ { x \rightarrow - 2 } \frac { x ^ { 3 } - x ^ { 2 } - 10 x - 8 } { - 2 x ^ { 3 } - 12 x ^ { 2 } - 6 x + 20 }$ exists. If the limit exists, estimate its value to two decimal places.

Accepted Answer

The answer of Use a graphing device to determine whether...

Question 11

Find the derivative of $f ( x ) = \frac { 1 } { \sqrt { x + 1 } }$ at the point $\left( 3 , \frac { 1 } { 2 } \right)$ .

Accepted Answer

None

Question 12

Find the derivative of the function at the given number. $g ( x ) = 2 - 2 x ^ { 2 }$ at 2

Accepted Answer

The answer of Find the derivative of the function at...

Question 13

Find $\lim _ { x \rightarrow \infty } \frac { 1 } { 2 x - 1 }$ .

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow \infty...

Question 14

Find the derivative of the function at the given number. $f ( x ) = 2 + 2 \sqrt { x }$ at 4

Accepted Answer

The answer of Find the derivative of the function at...

Question 15

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } \frac { 5 x } { \sqrt { 2 x ^ { 2 } + x } }$ . Then use a graphing device to confirm your result graphically.

Accepted Answer

The answer of Use a table of values to estimate...

Question 16

Determine whether the sequence $a _ { n } = \lim _ { n \rightarrow \infty } \frac { 15 } { n ^ { 3 } } \left[ \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \right]$ converges or diverges. If it converges, find the limit.

Accepted Answer

The answer of Determine whether the sequence \(a _ {...

Question 17

Evaluate the limit if it exists. $$\lim _ { t \rightarrow 0 } \left( \frac { 1 } { 3 t } - \frac { 1 } { t ^ { 2 } + 3 t } \right)$$

Accepted Answer

The answer of Evaluate the limit if it exists. \[\lim...

Question 18

Complete the table of values (to five decimal places) and use the table to estimate the value of $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { 9 - x }$ . $$\begin{array} { | l | l | l | l | l | l | l | } 
\hline x & 8.9 & 8.99 & 8.999 & 9.001 & 9.01 & 9.1 \
\hline f ( x ) & & & & & & \
\hline
\end{array}$$

Accepted Answer

None

Question 19

Estimate the area under the graph of $f ( x ) = x ^ { 2 } - 1$ from $x = 1$ to $x = 5$ using 
(a) four approximating rectangles and right endpoints.
(b) four approximating rectangles and left endpoints.
(c) eight approximating rectangles and right endpoints.

Accepted Answer

a) The rectangles have width @#LAT-DLM&

Question 20

Evaluate the limit if it exists. $$\lim _ { h \rightarrow 0 } \frac { ( 5 + h ) ^ { - 1 } - 5 ^ { - 1 } } { h }$$

Accepted Answer

The answer of Evaluate the limit if it exists. \[\lim...

Determine whether the sequence $a _ { \mathrm { n } } = \frac { n ^ { 4 } - 1 } { 5 - n ^ { 4 } }$ converges or diverges. If it converges, find the limit.

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 2 x$ over the interval $1 \leq x \leq 3$ .

Evaluate $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 4 x } { x - 4 }$ .

Evaluate $\lim _ { t \rightarrow 3 } \frac { t ^ { 2 } - 9 } { \frac { 1 } { 3 } - \frac { 1 } { t } }$ .

Find the derivative of $f ( x ) = \frac { 2 x } { x - 2 }$ at the point $\left( - 3 , \frac { \mathfrak { 6 } } {5 } \right)$ .

Find the derivative of $f ( x ) = 5 x - x ^ { 2 }$ at the point $\left( \frac { 5 } { 2 } , \frac { 25 } { 4 } \right)$ .

Evaluate $\lim _ { x \rightarrow 3/2 } \frac { 4 x ^ { 2 } - 16 x + 15 } { 2 x - 3 }$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 5$ over the interval $0 \leq x \leq 5$ .

Use a graphing device to determine whether $\lim _ { x \rightarrow - 2 } \frac { x ^ { 3 } - x ^ { 2 } - 10 x - 8 } { - 2 x ^ { 3 } - 12 x ^ { 2 } - 6 x + 20 }$ exists. If the limit exists, estimate its value to two decimal places.

Find the derivative of $f ( x ) = \frac { 1 } { \sqrt { x + 1 } }$ at the point $\left( 3 , \frac { 1 } { 2 } \right)$ .

Find the derivative of the function at the given number. $g ( x ) = 2 - 2 x ^ { 2 }$ at 2

Find $\lim _ { x \rightarrow \infty } \frac { 1 } { 2 x - 1 }$ .

Find the derivative of the function at the given number. $f ( x ) = 2 + 2 \sqrt { x }$ at 4

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } \frac { 5 x } { \sqrt { 2 x ^ { 2 } + x } }$ . Then use a graphing device to confirm your result graphically.

Determine whether the sequence $a _ { n } = \lim _ { n \rightarrow \infty } \frac { 15 } { n ^ { 3 } } \left[ \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \right]$ converges or diverges. If it converges, find the limit.

Evaluate the limit if it exists. $\lim _ { t \rightarrow 0 } \left( \frac { 1 } { 3 t } - \frac { 1 } { t ^ { 2 } + 3 t } \right)$

Complete the table of values (to five decimal places) and use the table to estimate the value of $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { 9 - x }$ . x 8.9 8.99 8.999 9.001 9.01 9.1 f(x)

Estimate the area under the graph of $f ( x ) = x ^ { 2 } - 1$ from $x = 1$ to $x = 5$ using (a) four approximating rectangles and right endpoints. (b) four approximating rectangles and left endpoints. (c) eight approximating rectangles and right endpoints.

Evaluate the limit if it exists. $\lim _ { h \rightarrow 0 } \frac { ( 5 + h ) ^ { - 1 } - 5 ^ { - 1 } } { h }$

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Filters

Exam 13: Limits: a Preview of Calculus

Determine whether the sequence an=n4−15−n4a _ { \mathrm { n } } = \frac { n ^ { 4 } - 1 } { 5 - n ^ { 4 } }an​=5−n4n4−1​ converges or diverges. If it converges, find the limit.

Use the definition of area as a limit to find the area of the region that lies under the graph of f(x)=2xf ( x ) = 2 xf(x)=2x over the interval 1≤x≤31 \leq x \leq 31≤x≤3 .

Evaluate lim⁡x→4x2−4xx−4\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 4 x } { x - 4 }limx→4​x−4x2−4x​ .

Evaluate lim⁡t→3t2−913−1t\lim _ { t \rightarrow 3 } \frac { t ^ { 2 } - 9 } { \frac { 1 } { 3 } - \frac { 1 } { t } }limt→3​31​−t1​t2−9​ .

Find the derivative of f(x)=2xx−2f ( x ) = \frac { 2 x } { x - 2 }f(x)=x−22x​ at the point (−3,65)\left( - 3 , \frac { \mathfrak { 6 } } {5 } \right)(−3,56​) .

Find the derivative of f(x)=5x−x2f ( x ) = 5 x - x ^ { 2 }f(x)=5x−x2 at the point (52,254)\left( \frac { 5 } { 2 } , \frac { 25 } { 4 } \right)(25​,425​) .

Evaluate lim⁡x→3/24x2−16x+152x−3\lim _ { x \rightarrow 3/2 } \frac { 4 x ^ { 2 } - 16 x + 15 } { 2 x - 3 }limx→3/2​2x−34x2−16x+15​ .

Use the definition of area as a limit to find the area of the region that lies under the graph of f(x)=5f ( x ) = 5f(x)=5 over the interval 0≤x≤50 \leq x \leq 50≤x≤5 .

Find the derivative of f(x)=1x+1f ( x ) = \frac { 1 } { \sqrt { x + 1 } }f(x)=x+1​1​ at the point (3,12)\left( 3 , \frac { 1 } { 2 } \right)(3,21​) .

Find the derivative of the function at the given number. g(x)=2−2x2g ( x ) = 2 - 2 x ^ { 2 }g(x)=2−2x2 at 2

Find lim⁡x→∞12x−1\lim _ { x \rightarrow \infty } \frac { 1 } { 2 x - 1 }limx→∞​2x−11​ .

Find the derivative of the function at the given number. f(x)=2+2xf ( x ) = 2 + 2 \sqrt { x }f(x)=2+2x​ at 4

Use a table of values to estimate the value of lim⁡x→∞5x2x2+x\lim _ { x \rightarrow \infty } \frac { 5 x } { \sqrt { 2 x ^ { 2 } + x } }limx→∞​2x2+x​5x​ . Then use a graphing device to confirm your result graphically.

Determine whether the sequence an=lim⁡n→∞15n3[n(n+1)(2n+1)6]a _ { n } = \lim _ { n \rightarrow \infty } \frac { 15 } { n ^ { 3 } } \left[ \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \right]an​=limn→∞​n315​[6n(n+1)(2n+1)​] converges or diverges. If it converges, find the limit.

Evaluate the limit if it exists. lim⁡t→0(13t−1t2+3t)\lim _ { t \rightarrow 0 } \left( \frac { 1 } { 3 t } - \frac { 1 } { t ^ { 2 } + 3 t } \right)limt→0​(3t1​−t2+3t1​)

Complete the table of values (to five decimal places) and use the table to estimate the value of lim⁡x→93−x9−x\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { 9 - x }limx→9​9−x3−x​​ . x 8.9 8.99 8.999 9.001 9.01 9.1 f(x)

Estimate the area under the graph of f(x)=x2−1f ( x ) = x ^ { 2 } - 1f(x)=x2−1 from x=1x = 1x=1 to x=5x = 5x=5 using (a) four approximating rectangles and right endpoints. (b) four approximating rectangles and left endpoints. (c) eight approximating rectangles and right endpoints.

Evaluate the limit if it exists. lim⁡h→0(5+h)−1−5−1h\lim _ { h \rightarrow 0 } \frac { ( 5 + h ) ^ { - 1 } - 5 ^ { - 1 } } { h }limh→0​h(5+h)−1−5−1​

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Filters

Determine whether the sequence $a _ { \mathrm { n } } = \frac { n ^ { 4 } - 1 } { 5 - n ^ { 4 } }$ converges or diverges. If it converges, find the limit.

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 2 x$ over the interval $1 \leq x \leq 3$ .

Evaluate $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 4 x } { x - 4 }$ .

Evaluate $\lim _ { t \rightarrow 3 } \frac { t ^ { 2 } - 9 } { \frac { 1 } { 3 } - \frac { 1 } { t } }$ .

Find the derivative of $f ( x ) = \frac { 2 x } { x - 2 }$ at the point $\left( - 3 , \frac { \mathfrak { 6 } } {5 } \right)$ .

Find the derivative of $f ( x ) = 5 x - x ^ { 2 }$ at the point $\left( \frac { 5 } { 2 } , \frac { 25 } { 4 } \right)$ .

Evaluate $\lim _ { x \rightarrow 3/2 } \frac { 4 x ^ { 2 } - 16 x + 15 } { 2 x - 3 }$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 5$ over the interval $0 \leq x \leq 5$ .

Find the derivative of $f ( x ) = \frac { 1 } { \sqrt { x + 1 } }$ at the point $\left( 3 , \frac { 1 } { 2 } \right)$ .

Find the derivative of the function at the given number. $g ( x ) = 2 - 2 x ^ { 2 }$ at 2

Find $\lim _ { x \rightarrow \infty } \frac { 1 } { 2 x - 1 }$ .

Find the derivative of the function at the given number. $f ( x ) = 2 + 2 \sqrt { x }$ at 4

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } \frac { 5 x } { \sqrt { 2 x ^ { 2 } + x } }$ . Then use a graphing device to confirm your result graphically.

Determine whether the sequence $a _ { n } = \lim _ { n \rightarrow \infty } \frac { 15 } { n ^ { 3 } } \left[ \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \right]$ converges or diverges. If it converges, find the limit.

Evaluate the limit if it exists. $\lim _ { t \rightarrow 0 } \left( \frac { 1 } { 3 t } - \frac { 1 } { t ^ { 2 } + 3 t } \right)$

Complete the table of values (to five decimal places) and use the table to estimate the value of $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { 9 - x }$ . x 8.9 8.99 8.999 9.001 9.01 9.1 f(x)

Estimate the area under the graph of $f ( x ) = x ^ { 2 } - 1$ from $x = 1$ to $x = 5$ using (a) four approximating rectangles and right endpoints. (b) four approximating rectangles and left endpoints. (c) eight approximating rectangles and right endpoints.

Evaluate the limit if it exists. $\lim _ { h \rightarrow 0 } \frac { ( 5 + h ) ^ { - 1 } - 5 ^ { - 1 } } { h }$