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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 15

Find the limit or show that it does not exist.

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

Stark L.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 30

Find the limit or show that it does not exist.

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

Ma. Theresa A.

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.

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.

Problem 33

Find the limit or show that it does not exist.

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

Suman Saurav T.

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.

Problem 35

Find the limit or show that it does not exist.

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

Stark L.

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.

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.

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.

Problem 39

Find the limit or show that it does not exist.

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

Leon D.

Problem 40

Find the limit or show that it does not exist.

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

Daniel J.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 58

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

Daniel J.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 77

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

Daniel J.

Problem 78

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

Daniel J.

Problem 79

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

Daniel J.

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.
(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 $$\lim_{x \to 0^+} x \sin \frac{1}{x}$$