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  • Calculus: Early Transcendentals
  • Limits and Derivatives

Calculus: Early Transcendentals

James Stewart

Chapter 2

Limits and Derivatives - all with Video Answers

Educators

+ 7 more educators

Section 6

Limits at Infinity: Horizontal Asymptotes

View

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  Alin
Ma. Theresa Alin
Numerade Educator
04:38

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.

Mary Wakumoto
Mary Wakumoto
Numerade Educator
View

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 Mccaslin
Numerade Educator
05:11

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 Qureshi
Umar Sohail Qureshi
Numerade Educator
03:54

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 Druch
Leon Druch
Numerade Educator
04:20

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 $

Mary Wakumoto
Mary Wakumoto
Numerade Educator
04:17

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 Kurse
Numerade Educator
09:00

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 Druch
Leon Druch
Numerade Educator
04:15

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 Druch
Leon Druch
Numerade Educator
06:35

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 Druch
Leon Druch
Numerade Educator
01:52

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 Jaimes
Daniel Jaimes
Numerade Educator
01:32

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 Jaimes
Daniel Jaimes
Numerade Educator
01:33

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 Han
Anthony Han
Numerade Educator
03:50

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 Jaimes
Daniel Jaimes
Numerade Educator
01:29

Problem 15

Find the limit or show that it does not exist.

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

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:37

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 Han
Anthony Han
Numerade Educator
01:25

Problem 17

Find the limit or show that it does not exist.

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

Anthony Han
Anthony Han
Numerade Educator
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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  Alin
Ma. Theresa Alin
Numerade Educator
02:16

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 Ledbetter
Stark Ledbetter
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
04:21

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 Druch
Leon Druch
Numerade Educator
01:37

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 Han
Anthony Han
Numerade Educator
02:53

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 Ledbetter
Stark Ledbetter
Numerade Educator
06:46

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 Cunningham
Evelyn Cunningham
Numerade Educator
04:19

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 Druch
Leon Druch
Numerade Educator
02:04

Problem 26

Find the limit or show that it does not exist.

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
02:23

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 Druch
Leon Druch
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
04:09

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 Hand
Linda Hand
Numerade Educator
View

Problem 30

Find the limit or show that it does not exist.

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

Ma. Theresa  Alin
Ma. Theresa Alin
Numerade Educator
02:39

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 Druch
Leon Druch
Numerade Educator
View

Problem 32

Find the limit or show that it does not exist.

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

Ma. Theresa  Alin
Ma. Theresa Alin
Numerade Educator
01:52

Problem 33

Find the limit or show that it does not exist.

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

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:47

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 Jaimes
Daniel Jaimes
Numerade Educator
01:52

Problem 35

Find the limit or show that it does not exist.

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

Stark Ledbetter
Stark Ledbetter
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
01:33

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 Ledbetter
Stark Ledbetter
Numerade Educator
03:08

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 Jaimes
Daniel Jaimes
Numerade Educator
07:25

Problem 39

Find the limit or show that it does not exist.

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

Leon Druch
Leon Druch
Numerade Educator
02:05

Problem 40

Find the limit or show that it does not exist.

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
08:25

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 Cooper
Michael Cooper
Numerade Educator
03:33

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 Jaimes
Daniel Jaimes
Numerade Educator
05:48

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 Jaimes
Daniel Jaimes
Numerade Educator
07:14

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 Jaimes
Daniel Jaimes
Numerade Educator
04:36

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 Druch
Leon Druch
Numerade Educator
07:28

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 Druch
Leon Druch
Numerade Educator
View

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  Alin
Ma. Theresa Alin
Numerade Educator
02:53

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 Jaimes
Daniel Jaimes
Numerade Educator
04:56

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 Jaimes
Daniel Jaimes
Numerade Educator
07:36

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 Druch
Leon Druch
Numerade Educator
03:54

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 Jaimes
Daniel Jaimes
Numerade Educator
06:06

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 Jaimes
Daniel Jaimes
Numerade Educator
05:41

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 Jaimes
Daniel Jaimes
Numerade Educator
07:13

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 Jaimes
Daniel Jaimes
Numerade Educator
07:39

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 Kurse
Numerade Educator
02:27

Problem 58

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
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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  Alin
Ma. Theresa Alin
Numerade Educator
05:41

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 Jaimes
Daniel Jaimes
Numerade Educator
04:18

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 Jaimes
Daniel Jaimes
Numerade Educator
04:42

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 Jaimes
Daniel Jaimes
Numerade Educator
05:39

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 Jaimes
Daniel Jaimes
Numerade Educator
07:01

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 Jaimes
Daniel Jaimes
Numerade Educator
03:27

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 Jaimes
Daniel Jaimes
Numerade Educator
04:02

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 Jaimes
Daniel Jaimes
Numerade Educator
03:53

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 Jaimes
Daniel Jaimes
Numerade Educator
02:44

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 Ledbetter
Stark Ledbetter
Numerade Educator
15:52

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 Sanchez
Eduard Sanchez
Numerade Educator
02:16

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 Jaimes
Daniel Jaimes
Numerade Educator
02:03

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 Jaimes
Daniel Jaimes
Numerade Educator
03:44

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 Jaimes
Daniel Jaimes
Numerade Educator
03:43

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 Jaimes
Daniel Jaimes
Numerade Educator
01:44

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 Jaimes
Daniel Jaimes
Numerade Educator
03:51

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 Jaimes
Daniel Jaimes
Numerade Educator
03:15

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 Jaimes
Daniel Jaimes
Numerade Educator
02:30

Problem 77

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
03:43

Problem 78

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
02:28

Problem 79

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

Daniel Jaimes
Daniel Jaimes
Numerade Educator
02:24

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 Jaimes
Daniel Jaimes
Numerade Educator
11:18

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 Jaimes
Daniel Jaimes
Numerade Educator

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