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Chapter 2

Limits and Derivatives

Educators

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+ 6 more educators

Problem 1

Explain in your own words the meaning of each of the following.

(a) $ \displaystyle \lim_{x \to \infty} f(x) = 5 $
(b) $ \displaystyle \lim_{x \to - \infty} f(x) = 3 $

Ma. Theresa A.
Numerade Educator

Problem 2

(a) Can the graph of $y=f(x)$ intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.
(b) How many horizontal asymptotes can the graph of $y=f(x)$ have? Sketch graphs to illustrate the possibilities.

DM
David M.
Numerade Educator

Problem 3

For the function $ f $ whose graph is given, state the following.
(a) $ \displaystyle \lim_{x \to \infty} f(x) $
(b) $ \displaystyle \lim_{x \to - \infty} f(x) $
(c) $ \displaystyle \lim_{x \to 1} f(x) $
(d) $ \displaystyle \lim_{x \to 3} f(x) $
(e) The equations of the asymptotes

DM
David M.
Numerade Educator

Problem 4

For the function $ g $ whose graph is given, state the following.

(a) $ \displaystyle \lim_{x \to \infty} g(x)$
(b) $ \displaystyle \lim_{x \to - \infty} g(x)$
(c) $ \displaystyle \lim_{x \to 0} g(x)$
(d) $ \displaystyle \lim_{x \to 2^-} g(x)$
(e) $ \displaystyle \lim_{x \to 2^+} g(x)$
(f) The equations of the asymptotes

Umar Sohail Q.
Numerade Educator

Problem 5

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 0} f(x) = -\infty $, $ \displaystyle \lim_{x \to -\infty} f(x) = 5 $, $ \displaystyle \lim_{x \to \infty} f(x) = -5 $

Leon D.
Numerade Educator

Problem 6

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 2} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^+} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^-} f(x) = -\infty $, $ \displaystyle \lim_{x \to -\infty} f(x) = 0 $, $ \displaystyle \lim_{x \to \infty} f(x) = 0 $, $ f(0) = 0 $

DR
Dhruv R.
Numerade Educator

Problem 7

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 2} f(x) = -\infty $, $ \displaystyle \lim_{x \to \infty} f(x) = \infty $, $ \displaystyle \lim_{x \to -\infty} f(x) = 0 $, $ \displaystyle \lim_{x \to 0^+} f(x) = \infty $, $ \displaystyle \lim_{x \to 0^-} f(x) = -\infty $,

AK
Anjali K.
Numerade Educator

Problem 8

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to \infty} f(x) = 3 $, $ \displaystyle \lim_{x \to 2^-} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^+} f(x) = -\infty $, $ f $ is odd

Leon D.
Numerade Educator

Problem 9

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ f(0) = 3 $, $ \displaystyle \lim_{x \to 0^-} f(x) = 4 $, $ \displaystyle \lim_{x \to 0^+} f(x) = 2 $, $ \displaystyle \lim_{x \to -\infty} f(x) = -\infty $, $ \displaystyle \lim_{x \to 4^-} f(x) = -\infty $, $ \displaystyle \lim_{x \to 4^+} f(x) = \infty $, $ \displaystyle \lim_{x \to \infty} f(x) = 3 $

Leon D.
Numerade Educator

Problem 10

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 3} f(x) = -\infty $, $ \displaystyle \lim_{x \to \infty} f(x) = 2 $, $ f(0) = 0 $, $ f $ is even

Leon D.
Numerade Educator

Problem 11

Guess the value of the limit $$ \lim_{x \to \infty} \frac{x^2}{2^x} $$ by evaluating the function $ f(x) = x^2/2^x $ for $ x $ = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of $ f $ to support your guess.

Daniel J.
Numerade Educator

Problem 12

(a) Use a graph of $$ f(x) = \left( 1 - \frac{2}{x} \right)^x $$ to estimate the value of $ \displaystyle \lim_{x \to \infty} f(x) $ correct to two decimal places.
(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

Daniel J.
Numerade Educator

Problem 13

Evaluate the limit and justify each step by indicating the appropriate properties of limits.

$ \displaystyle \lim_{x \to \infty} \dfrac{2x^2 - 7}{5x^2 + x -3} $

Anthony H.
Numerade Educator

Problem 14

Evaluate the limit and justify each step by indicating the appropriate properties of limits.

$ \displaystyle \lim_{x \to \infty} \sqrt{\dfrac{9x^3 + 8x - 4}{3 - 5x + x^3}} $

Daniel J.
Numerade Educator

Problem 15

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{3x - 2}{2x + 1} $

Stark L.
Numerade Educator

Problem 16

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{1 - x^2}{x^3 - x + 1} $

Anthony H.
Numerade Educator

Problem 17

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty}\frac{x - 2}{x^2 + 1} $

Anthony H.
Numerade Educator

Problem 18

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty}\frac{4x^3 + 6x^2 - 2}{2x^3 - 4x + 5} $

Ma. Theresa A.
Numerade Educator

Problem 19

Find the limit or show that it does not exist.

$ \displaystyle \lim_{t \to \infty}\frac{\sqrt{t} + t^2}{2t - t^2} $

Stark L.
Numerade Educator

Problem 20

Find the limit or show that it does not exist.

$ \displaystyle \lim_{t \to \infty}\frac{t - t\sqrt{t}}{2t^{3/2} + 3t - 5} $

Ma. Theresa A.
Numerade Educator

Problem 21

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{(2x^2 + 1)^2}{(x - 1)^2(x^2 + x)} $

Leon D.
Numerade Educator

Problem 22

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{x^2}{\sqrt{x^4 + 1}} $

Anthony H.
Numerade Educator

Problem 23

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{\sqrt{1 + 4x^6}}{2 - x^3} $

Stark L.
Numerade Educator

Problem 24

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty}\frac{\sqrt{1 + 4x^6}}{2 - x^3} $

Evelyn C.
Numerade Educator

Problem 25

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{\sqrt{x + 3x^2}}{4x - 1} $

Leon D.
Numerade Educator

Problem 26

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{x + 3x^2}{4x - 1} $

Daniel J.
Numerade Educator

Problem 27

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \left(\sqrt{9x^2 + x} - 3x \right) $

Leon D.
Numerade Educator

Problem 28

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty} \left(\sqrt{4x^2 + 3x} + 2x \right) $

Ma. Theresa A.
Numerade Educator

Problem 29

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \left(\sqrt{x^2 + ax} - \sqrt{x^2 + bx} \right) $

Linda H.
Numerade Educator

Problem 30

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \sqrt{x^2 + 1} $

Ma. Theresa A.
Numerade Educator

Problem 31

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{x^4 - 3x^2 + x}{x^3 - x + 2} $

Leon D.
Numerade Educator

Problem 32

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} (e^{-x} + 2 \cos 3x) $

Ma. Theresa A.
Numerade Educator

Problem 33

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty} (x^2 + 2x^7) $

Suman Saurav T.
Numerade Educator

Problem 34

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty} \frac{1 + x^6}{x^4 + 1} $

Daniel J.
Numerade Educator

Problem 35

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \arctan(e^x) $

Stark L.
Numerade Educator

Problem 36

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{e^{3x} - e^{-3x}}{e^{3x} + e^{-3x}} $

Ma. Theresa A.
Numerade Educator

Problem 37

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{1 - e^x}{1 + 2e^x} $

Stark L.
Numerade Educator

Problem 38

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{\sin^2 x}{x^2 + 1} $

Daniel J.
Numerade Educator

Problem 39

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} (e^{-2x}\cos x) $

Leon D.
Numerade Educator

Problem 40

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to 0^+} \tan^{-1}(\ln x) $

Daniel J.
Numerade Educator

Problem 41

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \bigl[\ln (1 + x^2) - \ln (1 + x) \bigr] $

Ma. Theresa A.
Numerade Educator

Problem 42

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \bigl[\ln (2 + x) - \ln (1 + x) \bigr] $

Ma. Theresa A.
Numerade Educator

Problem 43

(a) For $ f(x) = \frac{x}{\ln x} $ find each of the following limits.

(i) $ \displaystyle \lim_{x \to 0^+} f(x) $
(ii) $ \displaystyle \lim_{x \to 1^-} f(x) $
(iii) $ \displaystyle \lim_{x \to 1^+} f(x) $

(b) Use a table of values to estimate $ \displaystyle \lim_{x \to \infty} f(x) $.

(c) Use the information from parts (a) and (b) to make a rough sketch of the graph of $ f $.

Michael C.
Numerade Educator

Problem 44

For $ f(x) = \frac{2}{x} - \frac{1}{\ln x} $ find each of the following limits.

(a) $ \displaystyle \lim_{x \to \infty} f(x) $
(b) $ \displaystyle \lim_{x \to 0^+} f(x) $
(c) $ \displaystyle \lim_{x \to 1^-} f(x) $
(d) $ \displaystyle \lim_{x \to 1^+} f(x) $
(e) Use the information from parts (a) - (d) to make a rough sketch of the graph of $ f $.

Daniel J.
Numerade Educator

Problem 45

(a) Estimate the value of $$ \lim_{x \to -\infty} \left( \sqrt{x^2 + x + 1} + x \right) $$ by graphing the function $ f(x) = \sqrt{x^2 + x + 1} + x $.

(b) Use a table of values of $ f(x) $ to guess the value of the limit.

(c) Prove that your guess is correct.

Daniel J.
Numerade Educator

Problem 46

(a) Use a graph of $$ f(x) = \sqrt{3x^2 + 8x + 6} - \sqrt{3x^2 + 3x + 1} $$ to estimate the value of
$ \displaystyle \lim_{x \to \infty} f(x) $ to one decimal place.

(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

(c) Find the exact value of the limit.

Daniel J.
Numerade Educator

Problem 47

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{5 + 4x}{x + 3} $

Leon D.
Numerade Educator

Problem 48

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{2x^2 + 1}{3x^2 + 2x -1} $

Leon D.
Numerade Educator

Problem 49

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{2x^2 + x - 1}{x^2 + x -2} $

Ma. Theresa A.
Numerade Educator

Problem 50

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{1 + x^4}{x^2 - x^4} $

Daniel J.
Numerade Educator

Problem 51

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{x^3 - x}{x^2 - 6x + 5} $

Daniel J.
Numerade Educator

Problem 52

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{2e^x}{e^x - 5} $

Leon D.
Numerade Educator

Problem 53

Estimate the horizontal asymptote of the function
$$ f(x) = \frac{3x^3 + 500x^2}{x^3 + 500x^2 + 100x + 2000} $$
by graphing $ f $ for $ -10 \le x \le 10 $. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

Daniel J.
Numerade Educator

Problem 54

(a) Graph the function $$ f(x) = \frac{\sqrt{2x^2 + 1}}{3x - 5} $$
How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits $$ \lim_{x \to \infty} \frac{\sqrt{2x^2 + 1}}{3x - 5} \hspace{5mm} \text{and} \hspace{5mm} \lim_{x \to -\infty} \frac{\sqrt{2x^2 + 1}}{3x - 5} $$

(b) By calculating values of $ f(x) $, give numerical estimates of the limits in part (a).

(c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]

Daniel J.
Numerade Educator

Problem 55

Let $ P $ and $ Q $ be polynomials. Find $$ \lim_{x \to \infty} \frac{P(x)}{Q(x)} $$ if the degree of $ P $ is (a) less than the degree of $ Q $ and (b) greater than the degree of $ Q $.

Daniel J.
Numerade Educator

Problem 56

Make a rough sketch of the curve $ y = x^n $ ($ n $ an integer) for the following five cases:
(i) $ n = 0 $
(ii) $ n > 0 $, $ n $ odd
(iii) $ n > 0 $, $ n $ even
(iv) $ n < 0 $, $ n $ odd
(v) $ n < 0 $, $ n $ even
Then use these sketches to find the following limits.
(a) $ \displaystyle \lim_{x \to 0^+} x^n $
(b) $ \displaystyle \lim_{x \to 0^-} x^n $
(c) $ \displaystyle \lim_{x \to \infty} x^n $
(d) $ \displaystyle \lim_{x \to -\infty} x^n $

Daniel J.
Numerade Educator

Problem 57

Find a formula for a function $ f $ that satisfies the following conditions:
$$ \lim_{x \to \pm \infty} f(x) = 0, \lim_{x \to 0} f(x) = -\infty , f(2) = 0, \lim_{x \to 3^-} f(x) = \infty, \lim_{x \to 3^+} f(x) = -\infty $$

AK
Anjali K.
Numerade Educator

Problem 58

Find a formula for a function that has vertical asymptotes $ x = 1 $ and $ x = 3 $ and horizontal asymptote $ y = 1 $.

Daniel J.
Numerade Educator

Problem 59

A function $ f $ is a ratio of quadratic functions and has a vertical asymptote $ x = 4 $ and just one
$ x $-intercept, $ x = 1 $. It is known that $ f $ has a removable discontinuity at $ x = -1 $ and $ \displaystyle \lim_{x \to -1} f(x) = 2 $. Evaluate
(a) $ f (0) $ (b) $ \displaystyle \lim_{x \to \infty} f(x) $

Ma. Theresa A.
Numerade Educator

Problem 60

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = 2x^3 - x^4 $

Daniel J.
Numerade Educator

Problem 61

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = x^4 - x^6 $

Daniel J.
Numerade Educator

Problem 62

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = x^3(x + 2)^2(x - 1) $

Daniel J.
Numerade Educator

Problem 63

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = (3 - x)(1 + x)^2(1 - x)^4 $

Daniel J.
Numerade Educator

Problem 64

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = x^2(x^2 - 1)^2(x + 2) $

Daniel J.
Numerade Educator

Problem 65

(a) Use the Squeeze Theorem to evaluate $ \displaystyle \lim_{x \to \infty} \frac{\sin x}{x} $.
(b) Graph $ f(x) = (\sin x)/x $. How many times does the graph cross the asymptote?

Daniel J.
Numerade Educator

Problem 66

By the \textit{end behavior} of a function we mean the behavior of its values as $ x \to \infty $ and $ x \to -\infty $.
(a) Describe and compare the end behavior of the functions
$$ P(x) = 3x^5 - 5x^3 + 2x \hspace{5mm} Q(x) = 3x^5 $$
by graphing both functions in the viewing rectangles $ [-2, 2] $ by $ [-2, 2] $ and $ [-10, 10] $ by $ [-10,000, 10,000] $.
(b) Two functions are said to have the \textit{same end behavior} if their ratio approaches 1 as $ x \to \infty $. Show that $ P $ and $ Q $ have the same end behavior.

Daniel J.
Numerade Educator

Problem 67

Find $ \displaystyle \lim_{x \to \infty} f(x) $ if, for all $ x > 1 $, $$ \frac{10e^x - 21}{2e^x} < f(x) < \frac{5\sqrt{x}}{\sqrt{x - 1}} $$

Daniel J.
Numerade Educator

Problem 68

A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after $ t $ minutes (in grams per liter) is $$ C(t) = \frac{30t}{200 + t} $$

(b) What happens to the concentration as $ t \to \infty $?

Stark L.
Numerade Educator

Problem 69

In Chapter 9 we will be able to show, under certain assumptions, that the velocity $ v(t) $ of a falling raindrop at time $ t $ is $$ v(t) = v^*(1 - e^{-gt/v^*}) $$ where $ g $ is the acceleration due to gravity and $ v^* $ is the $terminal $ $ velocity $ of the raindrop.

(a) Find $ \displaystyle \lim_{t \to \infty} v(t) $.

(b) Graph $ v(t) $ if $ v^* = 1 m/s $ and $ g = 9.8 m/s^2 $. How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

Eduard S.
Numerade Educator

Problem 70

(a) By graphing $ y = e^{-x/10} $ and $ y = 0.1 $ on a common screen, discover how large you need to make $ x $ so that $ e^{-x/10} < 0.1 $.

(b) Can you solve part (a) without using a graphing device?

Daniel J.
Numerade Educator

Problem 71

Use a graph to find a number $ N $ such that
$$ \text{if} \hspace{5mm} x > N \hspace{5mm} \text{then} \hspace{5mm} \biggl| \frac{3x^2 + 1}{2x^2 + x + 1} - 1.5 \biggr| < 0.05 $$

Daniel J.
Numerade Educator

Problem 72

For the limit $$ \lim_{x \to \infty} \frac{1 - 3x}{\sqrt{x^2 + 1}} = -3 $$ illustrate Definition 7 by finding values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.05 $.

Daniel J.
Numerade Educator

Problem 73

For the limit $$ \lim_{x \to -\infty} \frac{1 - 3x}{\sqrt{x^2 + 1}} = 3 $$ illustrate Definition 8 by finding values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.05 $.

Daniel J.
Numerade Educator

Problem 74

For the limit $$ \lim_{x \to \infty} \sqrt{x \ln x} = \infty $$ illustrate Definition 9 by finding a value of $ N $ that corresponds to $ M = 100 $.

Daniel J.
Numerade Educator

Problem 75

(a) How large do we have to take $ x $ so that $ 1/x^2 < 0.0001 $?

(b) Take $ r = 2 $ in Theorem 5, we have the statement $$ \lim_{x \to \infty} \frac{1}{x^2} = 0 $$
Prove this directly using Definition 7.

Daniel J.
Numerade Educator

Problem 76

(a) How large do we have to take $ x $ so that $ 1/\sqrt{x} < 0.0001 $?

(b) Take $ r = \frac{1}{2} $ in Theorem 5, we have the statement
$$ \lim_{x \to \infty} \frac{1}{\sqrt{x}} = 0 $$
Prove this directly using Definition 7.

Daniel J.
Numerade Educator

Problem 77

Use Definition 8 to prove that $ \displaystyle \lim_{x \to -\infty} \frac{1}{x} = 0 $.

Daniel J.
Numerade Educator

Problem 78

Prove, using Definition 9, that $ \displaystyle \lim_{x \to \infty} x^3 = \infty $.

Daniel J.
Numerade Educator

Problem 79

Use Definition 9 to prove that $ \displaystyle \lim_{x \to \infty} e^x = \infty $.

Daniel J.
Numerade Educator

Problem 80

Formulate a precise definition of $$ \lim_{x \to -\infty} f(x) = -\infty $$ Then use your definition to prove that $$ \lim_{x \to -\infty} (1 + x^3) = -\infty $$

Daniel J.
Numerade Educator

Problem 81

(a) Prove that $ \displaystyle \lim_{x \to \infty} f(x) = \lim_{t \to 0^+} f(1/t) $ and $ \displaystyle \lim_{x \to -\infty} f(x) = \lim_{t \to 0^-} f(1/t) $ if these limits exist.

(b) Use part (a) and Exercise 65 to find $$ \lim_{x \to 0^+} x \sin \frac{1}{x} $$

Daniel J.
Numerade Educator