Question 1

Find the limit, if it exists.
-$$\lim _ { x \rightarrow -6 } \frac { x ^ { 2 } + 10 x + 24 } { x + 6 }$$

Accepted Answer

A)  $- 2$ 
B)  120 
C)  Does not exist 
D)  10 
A)  $- 2$ 
B)  120 
C)  Does not exist 
D)  10

Question 2

Prove the limit statement
-$$\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x - 2 } = 4$$

Accepted Answer

Let @#LAT-DLM& be given. Choose @#LAT-DLM&. Then @#LAT-DLM& implies th

Question 3

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-

Accepted Answer

A)  15 
B)  0.4 
C)  0.04 
D)  -0.04 
A)  15 
B)  0.4 
C)  0.04 
D)  -0.04

Question 4

Find the limit, if it exists.
-$\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }$

Accepted Answer

A)  Does not exist 
B)  $3 x ^ { 2 } + 3 x h + h ^ { 2 }$ 
C)  $3 x ^ { 2 }$ 
D)  0 
A)  Does not exist 
B)  $3 x ^ { 2 } + 3 x h + h ^ { 2 }$ 
C)  $3 x ^ { 2 }$ 
D)  0

Question 5

Use the graph to estimate the specified limit.
-Find lim x $\rightarrow$0 f(x)

Accepted Answer

A)  -1 
B)  1 
C)  does not exist 
D)  0 
A)  -1 
B)  1 
C)  does not exist 
D)  0

Question 6

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-f(x) = 5x + 2, L = 17, x0 = 3, and $\varepsilon$ = 0.01

Accepted Answer

A)  0.003333 
B)  0.004 
C)  0.002 
D)  0.01 
A)  0.003333 
B)  0.004 
C)  0.002 
D)  0.01

Question 7

Find the limit.
-$$\lim _ { x \rightarrow 0 } \sqrt { 15 + \cos ^ { 2 } x }$$

Accepted Answer

A)  15 
B)  $\sqrt { 15 }$ 
C)  4 
D)  16 
A)  15 
B)  $\sqrt { 15 }$ 
C)  4 
D)  16

Question 8

Determine the limit by sketching an appropriate graph.
-$$\lim _ { x \rightarrow 7 ^ { + } } f ( x ) \text {, where } f ( x ) = \left\{ \begin{array} { l l } 
x & - 7 \leq x  3
\end{array} \right.$$

Accepted Answer

A)  -7 
B)  -0 
C)  1 
D)  Does not exist 
A)  -7 
B)  -0 
C)  1 
D)  Does not exist

Question 9

Find the slope of the curve at the given point P and an equation of the tangent line at P.
-$y = x ^ { 3 } - 9 x , P ( 1 , - 8 )$

Accepted Answer

A)  slope is $- 6 ; y = - 6 x$ 
B)  slope is $3 ; y = 3 x - 11$ 
C)  slope is $3 ; y = 3 x - 7$ 
D)  slope is $- 6 ; y = - 6 x - 2$ 
A)  slope is $- 6 ; y = - 6 x$ 
B)  slope is $3 ; y = 3 x - 11$ 
C)  slope is $3 ; y = 3 x - 7$ 
D)  slope is $- 6 ; y = - 6 x - 2$

Question 10

Find the limit.
-If $\lim _ { x \rightarrow  1 } \frac { f ( x ) - 3 } { x - 1 } = 2$, find $\lim _ { x \rightarrow 1 } f ( x )$.

Accepted Answer

A)  1 
B)  2 
C)  3 
D)  Does not exist 
A)  1 
B)  2 
C)  3 
D)  Does not exist

Question 11

Give an appropriate answer.
-Let $\lim _ { x \rightarrow 9 } f ( x ) = 7$ and $\lim _ { x \rightarrow 9 } g ( x ) = - 10$. Find $\lim _ { x \rightarrow 9} \frac { f ( x ) } { g ( x ) }$.

Accepted Answer

A)  $- \frac { 10 } { 7 }$ 
B)  9 
C)  $- \frac { 7 } { 10 }$ 
D)  17 
A)  $- \frac { 10 } { 7 }$ 
B)  9 
C)  $- \frac { 7 } { 10 }$ 
D)  17

Question 12

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

Accepted Answer

A)  $- 4$ 
B)  $- 16$ 
C)  108 
D)  432 
A)  $- 4$ 
B)  $- 16$ 
C)  108 
D)  432

Question 13

Find the limit.
-If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 4$, find $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }$.

Accepted Answer

A)  2 
B)  8 
C)  16 
D)  4 
A)  2 
B)  8 
C)  16 
D)  4

Find the limit, if it exists. - $\lim _ { x \rightarrow -6 } \frac { x ^ { 2 } + 10 x + 24 } { x + 6 }$

Prove the limit statement - $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x - 2 } = 4$

Find the limit, if it exists. - $\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }$

Use the graph to estimate the specified limit. -Find lim x $\rightarrow$ 0 f(x) $Use the graph to estimate the specified limit. -Find lim x \rightarrow 0 f(x)$

Find the limit. - $\lim _ { x \rightarrow 0 } \sqrt { 15 + \cos ^ { 2 } x }$

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = x ^ { 3 } - 9 x , P ( 1 , - 8 )$

Find the limit. -If $\lim _ { x \rightarrow 1 } \frac { f ( x ) - 3 } { x - 1 } = 2$ , find $\lim _ { x \rightarrow 1 } f ( x )$ .

Give an appropriate answer. -Let $\lim _ { x \rightarrow 9 } f ( x ) = 7$ and $\lim _ { x \rightarrow 9 } g ( x ) = - 10$ . Find $\lim _ { x \rightarrow 9} \frac { f ( x ) } { g ( x ) }$ .

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

Find the limit. -If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 4$ , find $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }$ .

Functions

Differentiation

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

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Exam 2: Limits and Derivatives

Find the limit, if it exists. - lim⁡x→−6x2+10x+24x+6\lim _ { x \rightarrow -6 } \frac { x ^ { 2 } + 10 x + 24 } { x + 6 }limx→−6​x+6x2+10x+24​

Prove the limit statement - lim⁡x→2x2−4x−2=4\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x - 2 } = 4limx→2​x−2x2−4​=4

Find the limit, if it exists. - lim⁡h→0(x+h)3−x3h\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }limh→0​h(x+h)3−x3​

Use the graph to estimate the specified limit. -Find lim x →\rightarrow→ 0 f(x)

Find the limit. - lim⁡x→015+cos⁡2x\lim _ { x \rightarrow 0 } \sqrt { 15 + \cos ^ { 2 } x }limx→0​15+cos2x​

Find the slope of the curve at the given point P and an equation of the tangent line at P. - y=x3−9x,P(1,−8)y = x ^ { 3 } - 9 x , P ( 1 , - 8 )y=x3−9x,P(1,−8)

Find the limit. -If lim⁡x→1f(x)−3x−1=2\lim _ { x \rightarrow 1 } \frac { f ( x ) - 3 } { x - 1 } = 2limx→1​x−1f(x)−3​=2 , find lim⁡x→1f(x)\lim _ { x \rightarrow 1 } f ( x )limx→1​f(x) .

Give an appropriate answer. -Let lim⁡x→9f(x)=7\lim _ { x \rightarrow 9 } f ( x ) = 7limx→9​f(x)=7 and lim⁡x→9g(x)=−10\lim _ { x \rightarrow 9 } g ( x ) = - 10limx→9​g(x)=−10 . Find lim⁡x→9f(x)g(x)\lim _ { x \rightarrow 9} \frac { f ( x ) } { g ( x ) }limx→9​g(x)f(x)​ .

Find the limit if it exists. - lim⁡x→−1(x+3)2(x−2)3\lim _ { x \rightarrow - 1 } ( x + 3 ) ^ { 2 } ( x - 2 ) ^ { 3 }limx→−1​(x+3)2(x−2)3

Find the limit. -If lim⁡x→0f(x)x2=4\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 4limx→0​x2f(x)​=4 , find lim⁡x→0f(x)x\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }limx→0​xf(x)​ .