Deck 12: Limits, Derivatives, and Definite Integrals

Question 1

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{-}} f(x)$

Answer 1

3

Question 2

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} f(x)$

Answer 2

-1

Question 3

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3^{+}} f(x)$

Answer 3

approaches $-\infty$

Question 4

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-\infty} \frac{3 x-2}{2 x+3}$

Question 5

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3} \frac{x^{2}+x+4}{x^{2}-1}$

Question 6

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3} \frac{x^{2}-3 x}{x-3}$

Question 7

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3} \frac{x^{2}+4 x-21}{x-3}$

Question 8

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} \frac{3 x^{3}+7}{x^{4}+3 x^{2}+1}$

Question 9

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 0} \frac{x^{2}-4}{x^{3}+8}$

Question 10

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 0} \frac{\sin x}{4 x}$

Question 11

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-5} \frac{x^{2}+7 x+10}{x+5}$

Question 12

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 5} \sqrt{2 x^{2}-1}$

Question 13

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{+}} f(x)$ where $f(x)=\left\{\begin{array}{cc}2 x^{2}-7 & \text { if } x \leq 4 \\ 4-3 x & \text { if } x>4\end{array}\right.$

Question 14

$\lim _{x \rightarrow 8^{-}} 12$

Question 15

$\lim _{x \rightarrow 2^{-}}\left(3 x^{2}+2 x\right)$

Question 16

$\lim _{x \rightarrow 5^{+}} \sqrt{x-5}$

Question 17

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

Question 18

$\lim _{x \rightarrow 3^{+}} \sqrt{3-x}$

Question 19

Find the slope of the tangent line to the graph of $f(x)=5 x^{2}-3$ at the point where $x=1$ .

Question 20

Find the equation of the tangent line to the graph of $f(x)=\frac{2}{x}$ at the point $(1,2)$ .

Question 21

Use a calculator to estimate $f^{\prime}(2)$ , where $f(x)=\frac{2-3 e^{x}}{x}$ .

Question 22

Five grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 16 years?

Question 23

The total number of people who have seen a newly released movie at a particular theater chain follows the model $f(t)=10,000\left(1-e^{-.3 t}\right)$ , where $f(t)$ is the number of people who have seen the movie after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 16$ , and determine $f^{\prime}(5)$ . What does $f^{\prime}(5)$ tell about the situation?

Question 24

Approximate the area under the graph of $f(x)=2 x^{2}-1$ from $x=1$ to $x=5$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Question 25

Find $\int_{2}^{6}(2 x-4) d x$ by using the formula for the area of a triangle.

Question 26

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{+}} f(x)$

Question 27

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} f(x)$

Question 28

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3^{-}} f(x)$

Question 29

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-2} \frac{x^{2}+8 x+4}{8-x^{2}}$

Question 30

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 5} \sqrt{x^{2}-9}$

Question 31

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}}$

Question 32

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} \frac{6 x^{2}+5}{2 x^{2}-4 x+7}$

Question 33

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 2} \frac{x^{4}-2 x^{3}}{x-2}$

Question 34

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \frac{1}{3}} \frac{6 x^{2}+x-1}{3 x-1}$

Question 35

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 5} \frac{x^{2}-x-20}{x^{2}-25}$

Question 36

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-\infty} \frac{8 x-5}{9 x+5}$

Question 37

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 0} \frac{x^{3}+2 x-3}{x^{2}+5}$

Question 38

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3^{+}} f(x)$ where $f(x)= \begin{cases}x^{2}-2 x & \text { if } x<3 \\ 9-3 x & \text { if } x \geq 3\end{cases}$

Question 39

$\lim _{x \rightarrow 5^{-}} \sqrt{2 x-10}$

Question 40

$\lim _{x \rightarrow 3^{-}} \sqrt{9-x_{2}}$

Question 41

$\lim _{x \rightarrow 7^{-}} 11$

Question 42

$\lim _{x \rightarrow-1^{+}}\left(4 x^{2}-5 x+1\right)$

Question 43

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

Question 44

Find the slope of the tangent line to the graph of $f(x)=2 x^{2}+3 x$ at the point where $x=0$ .

Question 45

Find the equation of the tangent line to the graph of $f(x)=-\frac{6}{x}$ at the point $(2,-3)$ .

Question 46

Use a calculator to estimate $f^{\prime}(1)$ , where $f(x)=\frac{2-3 e^{x}}{3 x}$ .

Question 47

Five grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 20 years?

Question 48

During a two-week period the number of people in a town who have a virus follows the model $f(t)=15,000 e^{-.27 t}$ , where $f(t)$ is the number of people who have the virus after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 14$ , and determine $f^{\prime}(7)$ . What does $f^{\prime}(7)$ tell about the situation?

Question 49

Approximate the area under the graph of $f(x)=(x-4)^{2}+2$ from $x=0$ to $x=8$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Question 50

Find $\int_{-3}^{4}(x+3) d x$ by using the formula for the area of a triangle.

Question 51

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{-}} f(x)$

Question 52

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} f(x)$

Question 53

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 2^{+}} f(x)$

Question 54

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} \frac{1+5 x}{3 x+9}$

Question 55

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-1} \frac{x^{2}-2 x+4}{x^{2}-3}$

Question 56

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4} \frac{x^{2}-8 x+16}{x^{2}-16}$

Question 57

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4} \frac{x^{2}-10 x+24}{x-4}$

Question 58

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-\infty} \frac{5 x^{3}+1}{x^{2}+3 x-2}$

Question 59

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+4 x}$

Question 60

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x^{2}}$

Question 61

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3} \frac{x^{2}-3 x}{x-3}$

Question 62

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 3} \sqrt{16+x^{2}}$

Question 63

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 2^{-}} f(x)$ where $f(x)= \begin{cases}x^{2}+x & \text { if } x<2 \\ 3-5 x & \text { if } x \geq 2\end{cases}$

Question 64

$\lim _{x \rightarrow 2}\left(x^{3}+x^{2}+x+1\right)$

Question 65

$\lim _{x \rightarrow 200} 3$

Question 66

$\lim _{x \rightarrow-2^{+}} \sqrt{2 x+4}$

Question 67

$\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$

Question 68

$\lim _{x \rightarrow 0^{+}} \sqrt{2 x}$

Question 69

Find the slope of the tangent line to the graph of $f(x)=3 x^{2}-4$ at the point where $x=-2$ .

Question 70

Find the equation of the tangent line to the graph of $f(x)=\frac{10}{x}$ at the point $(-2,-5)$ .

Question 71

Use a calculator to estimate $f^{\prime}(1)$ , where $f(x)=\frac{e^{2 x}}{x}$ .

Question 72

Three grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 16 years?

Question 73

The population of bacteria in a culture follows the model $f(t)=6500\left(1-e^{-.35 t}\right)$ , where $f(t)$ is the number of bacteria after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 10$ , and determine $f^{\prime}(6)$ . What does $f^{\prime}(6)$ tell about the situation?

Question 74

Approximate the area under the graph of $f(x)=-x^{2}+9$ from $x=-1$ to $x=3$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Question 75

Find $\int_{0}^{5}(5-x) d x$ by using the formula for the area of a triangle.

Question 76

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{+}} f(x)$

Question 77

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow \infty} f(x)$

Question 78

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 4^{+}} f(x)$

Question 79

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 2} \frac{x^{2}-4 x+5}{3-x^{2}}$

Question 80

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so.
- $\lim _{x \rightarrow 9} \sqrt{x^{2}-17}$