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## Educators    ### Problem 1

Given that
$$\displaystyle \lim_{x\to a} f(x) = 0$$ $$\displaystyle \lim_{x\to a} g(x) = 0$$ $$\displaystyle \lim_{x\to a} h(x) = 1$$
$$\displaystyle \lim_{x\to a} p(x) = \infty$$ $$\displaystyle \lim_{x\to a} q(x) = \infty$$
which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible

(a) $\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}$
(b) $\displaystyle \lim_{x\to a} \frac{f(x)}{p(x)}$
(c) $\displaystyle \lim_{x\to a} \frac{h(x)}{p(x)}$
(d) $\displaystyle \lim_{x\to a} \frac{p(x)}{f(x)}$
(e) $\displaystyle \lim_{x\to a} \frac{p(x)}{q(x)}$ Carson M.

### Problem 2

Given that
$$\displaystyle \lim_{x\to a} f(x) = 0$$ $$\displaystyle \lim_{x\to a} g(x) = 0$$ $$\displaystyle \lim_{x\to a} h(x) = 1$$
$$\displaystyle \lim_{x\to a} p(x) = \infty$$ $$\displaystyle \lim_{x\to a} q(x) = \infty$$
which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible

(a) $$\displaystyle \lim_{x\to a} [f(x) p(x)]$$
(b) $$\displaystyle \lim_{x\to a} [h(x) p(x)]$$
(c) $$\displaystyle \lim_{x\to a} [p(x) q(x)]$$ Carson M.

### Problem 3

Given that
$$\displaystyle \lim_{x\to a} f(x) = 0$$ $$\displaystyle \lim_{x\to a} g(x) = 0$$ $$\displaystyle \lim_{x\to a} h(x) = 1$$
$$\displaystyle \lim_{x\to a} p(x) = \infty$$ $$\displaystyle \lim_{x\to a} q(x) = \infty$$
which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible

(a) $\displaystyle \lim_{x\to a} [f(x) - p(x)]$
(b) $\displaystyle \lim_{x\to a} [p(x) - q(x)]$
(c) $\displaystyle \lim_{x\to a} [p(x) + q(x)]$ Carson M.

### Problem 4

Given that
$$\displaystyle \lim_{x\to a} f(x) = 0$$ $$\displaystyle \lim_{x\to a} g(x) = 0$$ $$\displaystyle \lim_{x\to a} h(x) = 1$$
$$\displaystyle \lim_{x\to a} p(x) = \infty$$ $$\displaystyle \lim_{x\to a} q(x) = \infty$$
which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible

(a) $\displaystyle \lim_{x\to a} [f(x)]^{g(x)}$
(b) $\displaystyle \lim_{x\to a} [f(x)]^{p(x)}$
(c) $\displaystyle \lim_{x\to a} [h(x)]^{p(x)}$
(d) $\displaystyle \lim_{x\to a} [p(x)]^{f(x)}$
(e) $\displaystyle \lim_{x\to a} [p(x)]^{q(x)}$
(f) $\displaystyle \lim_{x\to a} \sqrt[q(x)]{p(x)}$ Nick J.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 J.

### 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 M.

### 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.

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

MY
Mengsha Y.

### 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 M.

### 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.

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

MY
Mengsha Y.

### 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 M.

### 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 M.

### 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 M.

### 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.

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

MY
Mengsha Y.

### 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 M.

### 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.

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

MY
Mengsha Y.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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.

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

MY
Mengsha Y.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 B.

### 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 B.

### 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 B.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 B.

### 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 M.

### 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 M.

### 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 M.

### 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 B.

### 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 B.

### 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 M.

### 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 M.

### 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 B.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### 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 M.

### Problem 72

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.

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

### 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 M.

### 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 B.

### 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 B.

### 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 B.

### 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 M.

### 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 M.

### 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 M.

### 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 A.

### 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 M.

### 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 M.

### 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
$$y = \frac{\sqrt{2a^3 x - x^4} - a\sqrt{aax}}{a - \sqrt{ax^3}}$$
as $x$ approaches $a$, where $a > 0$. (At that time it was common to write $aa$ instead of $a^2$.) Solve this problem. Carson M.

### 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 S.

### Problem 85

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

### 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 B.

### 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 M.

### 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 B.
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 M.

### Problem 90

If $f"$ is continuous, show that
$$\displaystyle \lim_{h\to 0} \frac{f(x + h) - 2f(x) + f(x - h)}{h^2} = f"(x)$$ Amrita B.

### Problem 91

Let
$f(x) = \left\{ \begin{array}{ll} e^{-1/x^2} & \mbox{if} x \not= 0\\ 0 & \mbox{if} x = 0\\ \end{array} \right.$
(a) Use the definition of derivative to compute $f'(0)$.
(b) Show that $f$ has derivatives of all orders that are defined on $\mathbb{R}$. [Hint: First show by induction that there is a polynomial $p_n(x)$ and a nonnegative integer $k_n$ such that $f^{(n)}(x) = p_n(x)f(x)/x^{k_n}$ for $x \not= 0$.] Carson M.
$$f(x) = \left\{ \begin{array}{ll} |x|^x & \mbox{if} x \not= 0\\ 1 & \mbox{if} x = 0\\ \end{array} \right.$$
(a) Show that $f$ is continuous at $0$.
(b) Investigate graphically whether $f$ is differentiable at $0$ by zooming in several times toward the point $(0, 1)$ on the graph of $f$.
(c) Show that $f$ is not differentiable at $0$. How can you reconcile this fact with the appearance of the graphs in part (b)? 