Calculus: Early Transcendentals

Problem 5

Use the graphs of $ f $ and $ g $ and their tangent lines at $ (2, 0) $ to find $ \displaystyle \lim_{x\to 2} \frac{f(x)}{g(x)} $.

Carson Merrill

Numerade Educator

01:43

Problem 6

Use the graphs of $ f $ and $ g $ and their tangent lines at $ (2, 0) $ to find $ \displaystyle \lim_{x\to 2} \frac{f(x)}{g(x)} $.

Carson Merrill

Numerade Educator

Problem 7

The graph of a function $ f $ and its tangent line at $ 0 $ are shown. What is the value of $ \displaystyle \lim_{x\to 0} \frac{f(x)}{e^x - 1} $?

Carson Merrill

Numerade Educator

01:40

Problem 8

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 3} \frac{x - 3}{x^2 - 9} $

Carson Merrill

Numerade Educator

Problem 9

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 4} \frac{x^2 - 2x - 8}{x - 4} $

Carson Merrill

Numerade Educator

01:11

Problem 10

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to -2} \frac{x^3 + 8}{x + 2} $

Carson Merrill

Numerade Educator

Problem 11

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Carson Merrill

Numerade Educator

01:23

Problem 12

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1/2} \frac{6x^2 + 5x - 4}{4x^2 + 16x - 9} $

Carson Merrill

Numerade Educator

01:37

Problem 13

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to (\pi /2)^+} \frac{\cos x}{1 - \sin x} $

Carson Merrill

Numerade Educator

01:30

Problem 14

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\tan 3x}{\sin 2x} $

Carson Merrill

Numerade Educator

Problem 15

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{t\to 0} \frac{e^{2t} - 1}{\sin t} $

Carson Merrill

Numerade Educator

01:13

Problem 16

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{x^2}{1 - \cos x} $

Carson Merrill

Numerade Educator

03:39

Problem 17

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{\theta \to \pi /2} \frac{1 - \sin \theta}{1 + \cos 2\theta} $

Nick Johnson

Numerade Educator

01:13

Problem 18

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{\theta \to \pi} \frac{1 + \cos \theta}{1 - \cos \theta} $

Carson Merrill

Numerade Educator

01:56

Problem 19

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \frac{\ln x}{\sqrt{x}} $

Mengsha Yao

Numerade Educator

01:06

Problem 20

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Carson Merrill

Numerade Educator

01:36

Problem 21

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} \frac{\ln x}{x} $

Mengsha Yao

Numerade Educator

01:43

Problem 22

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \theta} \frac{\ln \sqrt{x}}{x^2} $

Carson Merrill

Numerade Educator

01:25

Problem 23

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{t\to 1} \frac{t^8 - 1}{t^5 - 1} $

Carson Merrill

Numerade Educator

Problem 24

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{t\to 0} \frac{8^t - 5^t}{t} $

Carson Merrill

Numerade Educator

03:25

Problem 25

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\sqrt{1 + 2x} - \sqrt{1 - 4x}}{x} $

Mengsha Yao

Numerade Educator

01:48

Problem 26

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{\mu \to 0} \frac{e^{\mu /10}}{u^3} $

Carson Merrill

Numerade Educator

02:08

Problem 27

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{e^x - 1 - x}{x^2} $

Mengsha Yao

Numerade Educator

01:42

Problem 28

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\sinh x - x}{x^3} $

Carson Merrill

Numerade Educator

01:06

Problem 29

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\tanh x}{\tan x} $

Carson Merrill

Numerade Educator

01:46

Problem 30

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{x - \sin x}{x - \tan x} $

Carson Merrill

Numerade Educator

01:16

Problem 31

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\sin^{-1} x}{x} $

Carson Merrill

Numerade Educator

02:22

Problem 32

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \frac{(\ln x)^2}{x} $

Mengsha Yao

Numerade Educator

01:49

Problem 33

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{x3^x}{3^x - 1} $

Carson Merrill

Numerade Educator

01:48

Problem 34

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\cos mx - \cos nx}{x^2} $

Carson Merrill

Numerade Educator

01:13

Problem 35

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Carson Merrill

Numerade Educator

01:09

Problem 36

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \lim_{x\to 1} \frac{x\sin(x - 1)}{2x^2 - x - 1} $

Carson Merrill

Numerade Educator

01:14

Problem 37

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \lim_{x\to 0^+} \frac{\arctan(2x)}{\ln x} $

Carson Merrill

Numerade Educator

01:20

Problem 38

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \lim_{x\to 0^+} \frac{x^x - 1}{\ln x + x - 1} $

Carson Merrill

Numerade Educator

01:06

Problem 39

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \lim_{x\to 1} \frac{x^a - 1}{x^b - 1} $, $ b \not= 0 $

Carson Merrill

Numerade Educator

00:59

Problem 40

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \lim_{x\to 0} \frac{e^x - e^{-x} - 2x}{x - \sin x} $

Amrita Bhasin

Numerade Educator

01:00

Problem 41

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\cos x - 1 + \frac{1}{2}x^2}{x^4} $

Amrita Bhasin

Numerade Educator

01:02

Problem 42

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to a^+} \frac{\cos x \ln(x - a)}{\ln(e^x - e^a)} $

Amrita Bhasin

Numerade Educator

Problem 43

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x\sin(\pi /x) $

Carson Merrill

Numerade Educator

01:14

Problem 44

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \sqrt{xe^{-x/2}} $

Carson Merrill

Numerade Educator

Problem 45

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \sin 5x \csc 3x $

Carson Merrill

Numerade Educator

01:26

Problem 46

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to -\infty} x\ln \left( 1 - \frac{1}{x} \right) $

Carson Merrill

Numerade Educator

Problem 47

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x^3 e^{-x^2} $

Amrita Bhasin

Numerade Educator

02:11

Problem 48

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x^{3/2} \sin(1/x) $

Carson Merrill

Numerade Educator

01:32

Problem 49

Find the limit. Use l'Hospital's Rule where appropriate. if there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1^+} \ln x \tan(\pi x/2) $

Carson Merrill

Numerade Educator

01:04

Problem 50

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to (\pi/2)^-} \cos x \sec 5x $

Carson Merrill

Numerade Educator

00:34

Problem 51

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1} \left(\frac{x}{x - 1} - \frac{1}{\ln x} \right) $

Amrita Bhasin

Numerade Educator

00:55

Problem 52

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} (\csc x - \cot x) $

Amrita Bhasin

Numerade Educator

01:54

Problem 53

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} \left(\frac{1}{x} - \frac{1}{e^x - 1} \right) $

Carson Merrill

Numerade Educator

01:35

Problem 54

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Carson Merrill

Numerade Educator

00:48

Problem 55

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} (x - \ln x) $

Amrita Bhasin

Numerade Educator

01:08

Problem 56

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1^+} [\ln(x^7 - 1) - \ln(x^5 - 1)] $

Carson Merrill

Numerade Educator

Problem 57

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} x^{\sqrt{x}} $

Carson Merrill

Numerade Educator

01:39

Problem 58

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} (\tan 2x)^x $

Carson Merrill

Numerade Educator

01:31

Problem 59

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} (1 - 2x)^{1/x} $

Carson Merrill

Numerade Educator

Problem 60

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \left( 1 + \frac{a}{x} \right)^{bx} $

Carson Merrill

Numerade Educator

01:25

Problem 61

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1^+} x^{1/(1 - x)} $

Carson Merrill

Numerade Educator

01:16

Problem 62

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x^{(\ln 2)/(1 + \ln x)} $

Carson Merrill

Numerade Educator

01:32

Problem 63

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x^{1/x} $

Carson Merrill

Numerade Educator

Problem 64

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

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

Carson Merrill

Numerade Educator

01:24

Problem 65

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} (4x + 1)^{\cot x} $

Carson Merrill

Numerade Educator

01:27

Problem 66

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1} (2x - 1)^{\tan(\pi x/2)} $

Carson Merrill

Numerade Educator

Problem 67

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0^+} (1 + \sin 3x)^{1/x} $

Carson Merrill

Numerade Educator

01:49

Problem 68

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \left( \frac{2x - 3}{2x + 5} \right)^{2x + 1} $

Carson Merrill

Numerade Educator

01:59

Problem 69

Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value.

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

Carson Merrill

Numerade Educator

01:26

Problem 70

Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value.

$ \displaystyle \lim_{x\to 0} \frac{5^x - 4^x}{3^x - 2^x} $

Carson Merrill

Numerade Educator

01:22

Problem 71

Illustrate l'Hospital's Rule by graphing both $ f(x)/g(x) $ and $ f'(x)/g'(x) $ near $ x = 0 $ to see that these ratios have the same limit as $ x \to 0 $. Also, calculate the exact value of the limit.

$ f(x) = e^x - 1 $, $ g(x) = x^3 + 4x $

Carson Merrill

Numerade Educator

Problem 72

$ f(x) = 2x \sin x $, $ g(x) = \sec x - 1 $

Carson Merrill

Numerade Educator

04:47

Problem 73

Prove that
$$ \displaystyle \lim_{x\to \infty} \frac{e^x}{x^n} = \infty $$
for any positive integer $ n $. This shows that the exponential function approaches infinity faster than any power of $ x $.

Mutahar Mehkri

Numerade Educator

00:34

Problem 74

Prove that
$$ \displaystyle \lim_{x\to \infty} \frac{\ln x}{x^p} = 0 $$
for any number $ p > 0 $. This shows that the logarithmic function approaches infinity more slowly than any power of $ x $.

Amrita Bhasin

Numerade Educator

01:07

Problem 75

What happens if you try to use l'Hospital's Rule to find the limit? Evaluate the limit using another method.

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

Amrita Bhasin

Numerade Educator

01:04

Problem 76

What happens if you try to use l'Hospital's Rule to find the limit? Evaluate the limit using another method.

$ \displaystyle \lim_{x\to (\pi /2)^-} \frac{\sec x}{\tan x} $

Amrita Bhasin

Numerade Educator

Problem 77

Investigate the family of curves $ f(x) = e^x - cx $. In particular, find the limits as $ x \to \pm \infty $ and determine the values of $ c $ for which $ f $ has an absolute minimum. What happens to the minimum points as $ c $ increases?

Carson Merrill

Numerade Educator

01:23

Problem 78

If an object with mass $ m $ is dropped from rest, one model for its speed $ v $ after $ t $ seconds, taking air resistance into account, is
$$ v = \frac{mg}{c}(1 - e^{-ct/m}) $$
where $ g $ is the acceleration due to gravity and $ c $ is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; $ c $ is the proportionality constant.)
(a) Calculate $ lim_{t\to \infty} v $. What is the meaning of this limit?
(b) For fixed $ t $, use l'Hospital's Rule to calculate $ lim_{c\to 0^+} v $. What can you conclude about the velocity of a falling object in a vacuum?

Carson Merrill

Numerade Educator

01:36

Problem 79

If an initial amount $ A_0 $ of money is invested at an interest rate $ r $ compounded $ n $ times a year, the value of the investment after $ t $ years is
$$ A = A_0 \left( 1 + \frac{r}{n} \right)^{nt} $$
If we let $ n \to \infty $, we refer to the continuous compounding of interest. Use l'Hospital's Rule to show that if interest is compounded continuously, then the amount after $ t $ years is
$$ A = A_oe^{rt} $$

Carson Merrill

Numerade Educator

08:25

Problem 80

Light enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (see the figure.)

They detailed their findings of this phenomenon, known as the Stiles-Crawford effect of the first kind, in an important paper published in 1933. In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil as they expected. The percentage $ P $ of the total luminance entering a pupil of radius $ r mm $ that is sensed at the retina can be described by
$$ P = \frac{1 - 10^{-pr^2}}{pr^2 \ln 10} $$
where $ p $ is an experimentally determined constant, typically about $ 0.05 $.
(a) What is the percentage of luminance sensed by a pupil of radius $ 3 mm $? Use $ p = 0.05 $.
(b) Compute the percentage of luminance sensed by a pupil of radius $ 2 mm $. Does it make sense that it is larger than the answer to part $ (a) $?
(c) Compute $ \displaystyle \lim_{r\to 0^+} P $. Is the result what you would expect? Is this physically possible?

Source: Adapted from W. Stiles and B. Crawford, "The Luminous Efficiency of Ray Entering the Eye Pupil at Different Points." Proceedings of the Royal Society of London, Series B: Biological Sciences 112(1933): 428-50.

Donald Albin

Numerade Educator

Problem 81

Some populations initially grow exponentially but eventually level off. Equations of the form
$$ P(t) = \frac{M}{1 + Ae^{-kt}} $$
where $ M $, $ A $, and $ k $ are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here $ M $ is called the carrying capacity and represents the maximum population size that can be supported, and $ A = \frac{M - P_0}{P_0} $, where $ P_0 $ is the initial population.
(a) Compute $ lim_{t\to \infty} P(t) $. Explain why your answer is to be expected.
(b) Compute $ lim_{M\to \infty} P(t) $. (Note that $ A $ is defined in terms of $ M $.) What kind of function is your result?

Carson Merrill

Numerade Educator

01:44

Problem 82

A metal cable has radius $ r $ and is covered by insulation so that the distance from the center of the cable to the exterior of the insulation is $ R $. The velocity $ v $ of an electrical impulse in the cable is
$$ v = -c \left( \frac{r}{R} \right)^2 \ln \left( \frac{r}{R} \right) $$
where $ c $ is a positive constant. Find the following limits and interpret your answers.
(a) $ \displaystyle \lim_{R\to r^+}v $ (b) $ \displaystyle \lim_{r\to 0^+}v $

Carson Merrill

Numerade Educator

03:28

Problem 83

The first appearance in print of l'Hospital's Rule was in the book Analyse des Infiniment Petits published by the Marquis de l'Hospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function
$$ y = \frac{\sqrt{2a^3 x - x^4} - a\sqrt[3]{aax}}{a - \sqrt[4]{ax^3}} $$
as $ x $ approaches $ a $, where $ a > 0 $. (At that time it was common to write $ aa $ instead of $ a^2 $.) Solve this problem.

Carson Merrill

Numerade Educator

15:00

Problem 84

The figure shows a sector of a circle with central angle $ \theta $. Let $ A(\theta) $ be the area of the segment between the chord $ PR $ and the arc $ PR $. Let $ B(\theta) $ be the area of the triangle $ PQR $. Find the $ lim_{\theta\to 0^+} A(\theta)/B(\theta) $.

Aparna Shakti

Numerade Educator

01:23

Problem 85

Evaluate
$$ \displaystyle \lim_{x\to \infty} \left[ x - x^2 \ln \left( \frac{1 + x}{x} \right) \right] $$.

Carson Merrill

Numerade Educator

Problem 86

Suppose $ f $ is a positive function. If $ lim_{x\to a} f(x) = 0 $ and $ lim_{x\to a} g(x) = \infty $, show that
$$ \displaystyle \lim_{x\to a} [f(x)]^{g(x)} = 0 $$
This shows that $ 0^{\infty} $ is not an indeterminate form.

Amrita Bhasin

Numerade Educator

01:24

Problem 87

If $ f' $ is continuous, $ f(2) = 0 $, and $ f'(2) = 7 $, evaluate
$$ \displaystyle \lim_{x\to 0} \frac{f(2 + 3x) + f(2 + 5x)}{x} $$

Carson Merrill

Numerade Educator

06:49

Problem 88

For what values of $ a $ and $ b $ is the following equation true?
$$ \displaystyle \lim_{x\to 0} \left( \frac{\sin 2^x}{x^3} + a + \frac{b}{x^2} \right) = 0 $$

Bobby Barnes

University of North Texas

Problem 89

If $ f' $ is continuous, use l'Hospital's Rule to show that
$$ \displaystyle \lim_{h\to 0} \frac{f(x + h) - f(x - h)}{2h} = f'(x) $$
Explain the meaning of this equation with the aid of a diagram.

Carson Merrill

Numerade Educator