Exam 13: Limits: a Preview of Calculus

For the function $f$ whose graph is given, state the value of the given quantity. A) $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ B) $\lim _ { x \rightarrow - 2 ^ { + } } f ( x )$ C) $\lim _ { x \rightarrow - 2 } f ( x )$ D) $f ( - 2 )$ $For the function f whose graph is given, state the value of the given quantity. A) \lim _ { x \rightarrow - 2 ^ { - } } f ( x ) B) \lim _ { x \rightarrow - 2 ^ { + } } f ( x ) C) \lim _ { x \rightarrow - 2 } f ( x ) D) f ( - 2 )$

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Question 41

Use a table of values to estimate the value of $\lim _ { x \rightarrow 0 } \frac { 4 ^ { x } + 2 ^ { x } - 2 } { x }$ . Use a graphing device to confirm your answer.

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Question 42

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 3 } } { 2 }$ at the point $\left( 1 , \frac { 1 } { 2 } \right)$ .

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Question 43

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

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Question 44

Evaluate $\lim _ { x \rightarrow - 2 } ( x + 1 ) ^ { 5 } \sqrt { 2 x ^ { 4 } + x ^ { 2 } }$ , and justify each step by indicating the appropriate Limit Law(s).

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Question 45

Find the value of $\lim _ { x \rightarrow 1 ^ { + } } \frac { | 1 - x | } { x - x ^ { 2 } }$ , if it exists. If the limit does not exist, explain why.

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Question 46

If $f ( x ) = \frac { x + 1 } { x }$ , find $f ^ { \prime } ( a )$ .

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Question 47

If $f ( x ) = \frac { 2 } { \sqrt { x } }$ , find $f ^ { \prime } ( a )$ .

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Question 48

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 4 x ^ { 3 } + 1$ over the interval $0 \leq x \leq 6$ .

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Question 49

Determine whether the sequence $a _ { n } = \cos ( n ( 1 + 2 \pi ) )$ converges or diverges. If it converges, find the limit.

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Question 50

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } ( \sqrt { x + 1 } - \sqrt { x - 1 } )$ . Then use a graphing device to confirm your result graphically.

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Question 51

A particle moves in a straight line with its displacement of motion described by equation $s ( t ) = t ^ { 2 } - 3 t + 5$ , where $s$ is measured in feet and $t$ is measured in seconds. Find the velocity of $s$ at $t = a$ , $t = 2$ , $t = 4$ , and $t = 6$ .

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Question 52

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 3 x + 1$ over the interval $0 \leq x \leq 4$ .

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Question 53

Use a table of values to estimate the value of $\lim _ { x \rightarrow - ] } \frac { 81 - x ^ { 4 } } { 3 + x }$ . Use a graphing device to confirm your answer graphically.

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Question 54

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 2 } } { 2 } + 1$ at the point $( 4,9 )$ .

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Question 55

Find the equation of the tangent line to the curve $y =$ $x - 2 x ^ { 2 }$ at the point $( 2 , - 6 )$ .

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Question 56

For the function g whose graph is given, state the value of the given quantity if it exists. $For the function g whose graph is given, state the value of the given quantity if it exists. a) \lim _ { x \rightarrow 0 } g ( t ) b) g ( 0 )$ a) $\lim _ { x \rightarrow 0 } g ( t )$ b) $g ( 0 )$

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Question 57

Determine whether the sequence $\alpha _ { n } = \frac { 1 } { e ^ { n } }$ converges or diverges. If it converges, find the limit.

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Question 58

A stone is dropped into a pond causing a circular ripple in the water. Find the rate of change of the area $A$ of the circle with respect to the radius when $r= 3.5$ ft.

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Question 59

Find $\lim _ { x \rightarrow \infty } \frac { 5 - 4 x } { 4 x - 5 }$ .

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Question 60

Use a table of values to estimate the value of $\lim _ { x \rightarrow 0 } \frac { 4 ^ { x } + 2 ^ { x } - 2 } { x }$ . Use a graphing device to confirm your answer.

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 3 } } { 2 }$ at the point $\left( 1 , \frac { 1 } { 2 } \right)$ .

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

Evaluate $\lim _ { x \rightarrow - 2 } ( x + 1 ) ^ { 5 } \sqrt { 2 x ^ { 4 } + x ^ { 2 } }$ , and justify each step by indicating the appropriate Limit Law(s).

Find the value of $\lim _ { x \rightarrow 1 ^ { + } } \frac { | 1 - x | } { x - x ^ { 2 } }$ , if it exists. If the limit does not exist, explain why.

If $f ( x ) = \frac { x + 1 } { x }$ , find $f ^ { \prime } ( a )$ .

If $f ( x ) = \frac { 2 } { \sqrt { x } }$ , find $f ^ { \prime } ( a )$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 4 x ^ { 3 } + 1$ over the interval $0 \leq x \leq 6$ .

Determine whether the sequence $a _ { n } = \cos ( n ( 1 + 2 \pi ) )$ converges or diverges. If it converges, find the limit.

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } ( \sqrt { x + 1 } - \sqrt { x - 1 } )$ . Then use a graphing device to confirm your result graphically.

A particle moves in a straight line with its displacement of motion described by equation $s ( t ) = t ^ { 2 } - 3 t + 5$ , where $s$ is measured in feet and $t$ is measured in seconds. Find the velocity of $s$ at $t = a$ , $t = 2$ , $t = 4$ , and $t = 6$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 3 x + 1$ over the interval $0 \leq x \leq 4$ .

Use a table of values to estimate the value of $\lim _ { x \rightarrow - ] } \frac { 81 - x ^ { 4 } } { 3 + x }$ . Use a graphing device to confirm your answer graphically.

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 2 } } { 2 } + 1$ at the point $( 4,9 )$ .

Find the equation of the tangent line to the curve $y =$ $x - 2 x ^ { 2 }$ at the point $( 2 , - 6 )$ .

For the function g whose graph is given, state the value of the given quantity if it exists. $For the function g whose graph is given, state the value of the given quantity if it exists. a) \lim _ { x \rightarrow 0 } g ( t ) b) g ( 0 )$ a) $\lim _ { x \rightarrow 0 } g ( t )$ b) $g ( 0 )$

Determine whether the sequence $\alpha _ { n } = \frac { 1 } { e ^ { n } }$ converges or diverges. If it converges, find the limit.

A stone is dropped into a pond causing a circular ripple in the water. Find the rate of change of the area $A$ of the circle with respect to the radius when $r= 3.5$ ft.

Find $\lim _ { x \rightarrow \infty } \frac { 5 - 4 x } { 4 x - 5 }$ .

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

Use a table of values to estimate the value of lim⁡x→04x+2x−2x\lim _ { x \rightarrow 0 } \frac { 4 ^ { x } + 2 ^ { x } - 2 } { x }limx→0​x4x+2x−2​ . Use a graphing device to confirm your answer.

Find the slope of the tangent line of the graph of f(x)=x32f ( x ) = \frac { x ^ { 3 } } { 2 }f(x)=2x3​ at the point (1,12)\left( 1 , \frac { 1 } { 2 } \right)(1,21​) .

Determine whether the sequence an=(−1)n.1nn+1a _ { n } = \frac { ( - 1 ) ^ { n.1 } n } { n + 1 }an​=n+1(−1)n.1n​ converges or diverges. If it converges, find the limit.

Evaluate lim⁡x→−2(x+1)52x4+x2\lim _ { x \rightarrow - 2 } ( x + 1 ) ^ { 5 } \sqrt { 2 x ^ { 4 } + x ^ { 2 } }limx→−2​(x+1)52x4+x2​ , and justify each step by indicating the appropriate Limit Law(s).

Find the value of lim⁡x→1+∣1−x∣x−x2\lim _ { x \rightarrow 1 ^ { + } } \frac { | 1 - x | } { x - x ^ { 2 } }limx→1+​x−x2∣1−x∣​ , if it exists. If the limit does not exist, explain why.

If f(x)=x+1xf ( x ) = \frac { x + 1 } { x }f(x)=xx+1​ , find f′(a)f ^ { \prime } ( a )f′(a) .

If f(x)=2xf ( x ) = \frac { 2 } { \sqrt { x } }f(x)=x​2​ , find f′(a)f ^ { \prime } ( a )f′(a) .

Use the definition of area as a limit to find the area of the region that lies under the graph of f(x)=4x3+1f ( x ) = 4 x ^ { 3 } + 1f(x)=4x3+1 over the interval 0≤x≤60 \leq x \leq 60≤x≤6 .

Determine whether the sequence an=cos⁡(n(1+2π))a _ { n } = \cos ( n ( 1 + 2 \pi ) )an​=cos(n(1+2π)) converges or diverges. If it converges, find the limit.

Use a table of values to estimate the value of lim⁡x→∞(x+1−x−1)\lim _ { x \rightarrow \infty } ( \sqrt { x + 1 } - \sqrt { x - 1 } )limx→∞​(x+1​−x−1​) . Then use a graphing device to confirm your result graphically.

A particle moves in a straight line with its displacement of motion described by equation s(t)=t2−3t+5s ( t ) = t ^ { 2 } - 3 t + 5s(t)=t2−3t+5 , where sss is measured in feet and ttt is measured in seconds. Find the velocity of sss at t=at = at=a , t=2t = 2t=2 , t=4t = 4t=4 , and t=6t = 6t=6 .

Use the definition of area as a limit to find the area of the region that lies under the graph of f(x)=3x+1f ( x ) = 3 x + 1f(x)=3x+1 over the interval 0≤x≤40 \leq x \leq 40≤x≤4 .

Use a table of values to estimate the value of lim⁡x→−]81−x43+x\lim _ { x \rightarrow - ] } \frac { 81 - x ^ { 4 } } { 3 + x }limx→−]​3+x81−x4​ . Use a graphing device to confirm your answer graphically.

Find the slope of the tangent line of the graph of f(x)=x22+1f ( x ) = \frac { x ^ { 2 } } { 2 } + 1f(x)=2x2​+1 at the point (4,9)( 4,9 )(4,9) .

Find the equation of the tangent line to the curve y=y =y= x−2x2x - 2 x ^ { 2 }x−2x2 at the point (2,−6)( 2 , - 6 )(2,−6) .

For the function g whose graph is given, state the value of the given quantity if it exists. a) lim⁡x→0g(t)\lim _ { x \rightarrow 0 } g ( t )limx→0​g(t) b) g(0)g ( 0 )g(0)

Determine whether the sequence αn=1en\alpha _ { n } = \frac { 1 } { e ^ { n } }αn​=en1​ converges or diverges. If it converges, find the limit.

A stone is dropped into a pond causing a circular ripple in the water. Find the rate of change of the area AAA of the circle with respect to the radius when r=3.5r= 3.5r=3.5 ft.

Find lim⁡x→∞5−4x4x−5\lim _ { x \rightarrow \infty } \frac { 5 - 4 x } { 4 x - 5 }limx→∞​4x−55−4x​ .

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

Use a table of values to estimate the value of $\lim _ { x \rightarrow 0 } \frac { 4 ^ { x } + 2 ^ { x } - 2 } { x }$ . Use a graphing device to confirm your answer.

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 3 } } { 2 }$ at the point $\left( 1 , \frac { 1 } { 2 } \right)$ .

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

Evaluate $\lim _ { x \rightarrow - 2 } ( x + 1 ) ^ { 5 } \sqrt { 2 x ^ { 4 } + x ^ { 2 } }$ , and justify each step by indicating the appropriate Limit Law(s).

Find the value of $\lim _ { x \rightarrow 1 ^ { + } } \frac { | 1 - x | } { x - x ^ { 2 } }$ , if it exists. If the limit does not exist, explain why.

If $f ( x ) = \frac { x + 1 } { x }$ , find $f ^ { \prime } ( a )$ .

If $f ( x ) = \frac { 2 } { \sqrt { x } }$ , find $f ^ { \prime } ( a )$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 4 x ^ { 3 } + 1$ over the interval $0 \leq x \leq 6$ .

Determine whether the sequence $a _ { n } = \cos ( n ( 1 + 2 \pi ) )$ converges or diverges. If it converges, find the limit.

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } ( \sqrt { x + 1 } - \sqrt { x - 1 } )$ . Then use a graphing device to confirm your result graphically.

A particle moves in a straight line with its displacement of motion described by equation $s ( t ) = t ^ { 2 } - 3 t + 5$ , where $s$ is measured in feet and $t$ is measured in seconds. Find the velocity of $s$ at $t = a$ , $t = 2$ , $t = 4$ , and $t = 6$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 3 x + 1$ over the interval $0 \leq x \leq 4$ .

Use a table of values to estimate the value of $\lim _ { x \rightarrow - ] } \frac { 81 - x ^ { 4 } } { 3 + x }$ . Use a graphing device to confirm your answer graphically.

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 2 } } { 2 } + 1$ at the point $( 4,9 )$ .

Find the equation of the tangent line to the curve $y =$ $x - 2 x ^ { 2 }$ at the point $( 2 , - 6 )$ .

For the function g whose graph is given, state the value of the given quantity if it exists. $For the function g whose graph is given, state the value of the given quantity if it exists. a) \lim _ { x \rightarrow 0 } g ( t ) b) g ( 0 )$ a) $\lim _ { x \rightarrow 0 } g ( t )$ b) $g ( 0 )$

Determine whether the sequence $\alpha _ { n } = \frac { 1 } { e ^ { n } }$ converges or diverges. If it converges, find the limit.

A stone is dropped into a pond causing a circular ripple in the water. Find the rate of change of the area $A$ of the circle with respect to the radius when $r= 3.5$ ft.

Find $\lim _ { x \rightarrow \infty } \frac { 5 - 4 x } { 4 x - 5 }$ .