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Chapter 2

Limits and Derivatives

Educators

AK
+ 63 more educators

Problem 1

Explain in your own words what is meant by the equation $ \displaystyle \lim_{x\to 2} f(x) = 5 $
Is it possible for this statement to be true and yet $ f(2) = 3 $?
Explain.

AK
Anjali K.
Numerade Educator

Problem 2

Explain what it means to say that
$ \displaystyle \lim_{x\to1^-}f(x) = 3 $ and $ \displaystyle \lim_{x\to1^+}f(x) = 7 $

In this situation is it possible that $ \displaystyle \lim_{x\to1}f(x) $ exists?
Explain.

Dakarai H.
Numerade Educator

Problem 3

Explain the meaning of each of the following.
(a) $ \displaystyle \lim_{x\to-3}f(x) = \infty $ (b) $ \displaystyle \lim_{x\to4^+}f(x) = - \infty $

Ma. Theresa A.
Numerade Educator

Problem 4

Use the given graph of $ f $ to state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $ \displaystyle \lim_{x\to 2^-}f(x) $
(b) $ \displaystyle \lim_{x\to 2^+}f(x) $
(c) $ \displaystyle \lim_{x\to 2}f(x) $
(d) $ f(2) $
(e) $ \displaystyle \lim_{x\to 4}f(x) $
(f) $ f(4) $

Aparna S.
Numerade Educator

Problem 5

For the function $ f $ whose graph is given, state the value of each quantity, if it exists. If it does not, explain why.

(a) $ \displaystyle \lim_{x\to 1}f(x) $
(b) $ \displaystyle \lim_{x\to 3^-}f(x) $
(c) $ \displaystyle \lim_{x\to 3^+}f(x) $
(d) $ \displaystyle \lim_{x\to 3}f(x) $
(e) $ f(3) $

DM
David M.
Numerade Educator

Problem 6

For the function $ h $ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $ \displaystyle \lim_{x\to -3^-}h(x) $
(b) $ \displaystyle \lim_{x\to -3^+}h(x) $
(c) $ \displaystyle \lim_{x\to -3}h(x) $
(d) $ h(-3) $
(e) $ \displaystyle \lim_{x\to 0^-}h(x) $
(f) $ \displaystyle \lim_{x\to 0^+}h(x) $
(g) $ \displaystyle \lim_{x\to 0}h(x) $
(h) $ h(0) $
(i) $ \displaystyle \lim_{x\to 2}h(x) $
(j) $ h(2) $
(k) $ \displaystyle \lim_{x\to 5^+}h(x) $
(l) $ \displaystyle \lim_{x\to 5^-}h(x) $

DM
David M.
Numerade Educator

Problem 7

For the function $ g $ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $ \displaystyle \lim_{t\to 0^-}g(t) $
(b) $ \displaystyle \lim_{t\to 0^+}g(t) $
(c) $ \displaystyle \lim_{t\to 0}g(t) $
(d) $ \displaystyle \lim_{t\to 2^-}g(t) $
(e) $ \displaystyle \lim_{t\to 2^+}g(t) $
(f) $ \displaystyle \lim_{t\to 2}g(t) $
(g) $ g(2) $
(h) $ \displaystyle \lim_{t\to 4}g(t) $

DM
David M.
Numerade Educator

Problem 8

For the function $ A $ whose graph is shown, state the following.

(a) $ \displaystyle \lim_{x\to -3}A(x) $
(b) $ \displaystyle \lim_{x\to 2^-}A(x) $
(c) $ \displaystyle \lim_{x\to 2^+}A(x) $
(d) $ \displaystyle \lim_{x\to -1}A(x) $
(e) The equations of the vertical asymptotes

DM
David M.
Numerade Educator

Problem 9

For the function $ f $ whose graph is shown, state the following.

(a) $ \displaystyle \lim_{x\to -7}f(x) $
(b) $ \displaystyle \lim_{x\to -3}f(x) $
(c) $ \displaystyle \lim_{x\to 0}f(x) $
(d) $ \displaystyle \lim_{x\to 6^-}f(x) $
(e) $ \displaystyle \lim_{x\to 6^+}f(x) $
(f) The equations of the vertical asymptotes.

Aparna S.
Numerade Educator

Problem 10

A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount $ f(t) $ of the drug in the bloodstream after $ t $ hours. Find
$ \displaystyle \lim_{t\to 12^-}f(t) $ and $ \displaystyle \lim_{t\to 12^+}f(t) $
and explain the significance of these one-sided limits.

DM
David M.
Numerade Educator

Problem 11

Sketch the graph of the function and use it to determine the values of $ a $ for which $ \displaystyle \lim_{x\to a}f(x) $ exists.
$$ f(x) = \left\{
\begin{array}{ll}
1 + x & \mbox {if $ x < -1 $}\\
x^2 & \mbox{if $ -1 \le x < 1$}\\
2 - x & \mbox{if $ x \ge 1 $}
\end{array} \right.$$

Leon D.
Numerade Educator

Problem 12

Sketch the graph of the function and use it to determine the values of $ a $ for which $ \displaystyle \lim_{x\to a}f(x) $ exists.
$ f(x) = \left\{
\begin{array}{ll}
1 + \sin x & \mbox{if $ x < 0 $}\\
\cos x & \mbox{if $ 0 \le x \le \pi $}\\
\sin x & \mbox{if $ x > \pi $}
\end{array} \right.$

Leon D.
Numerade Educator

Problem 13

Use the graph of the function $ f $ to state the value of each limit, if it exists. If it does not exist, explain why.
(a) $ \displaystyle \lim_{x \to 0^-}f(x) $
(b) $ \displaystyle \lim_{x \to 0^+}f(x) $
(c) $ \displaystyle \lim_{x \to 0}f(x) $

$ \displaystyle f(x) = \frac{1}{1+e^{1/x}} $

DM
David M.
Numerade Educator

Problem 14

Use the graph of the function $ f $ to state the value of each limit, if it exists. If it does not exist, explain why.
(a) $ \displaystyle \lim_{x \to 0^-}f(x) $
(b) $ \displaystyle \lim_{x \to 0^+}f(x) $
(c) $ \displaystyle \lim_{x \to 0}f(x) $

$ \displaystyle f(x) = \frac{x^2 + x}{\sqrt{x^3 + x^2}} $

Daniel J.
Numerade Educator

Problem 15

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

$ \displaystyle \lim_{x \to 0^-}f(x) = -1 $, $ \displaystyle \lim_{x \to 0^+}f(x) = 2 $, $ f(0) = 1 $

Stark L.
Numerade Educator

Problem 16

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

$ \displaystyle \lim_{x \to 0}f(x) = 1 $, $ \displaystyle \lim_{x \to 3^-}f(x) = -2 $, $ \displaystyle \lim_{x \to 3^+}f(x) = 2 $, $ f(0) = -1 $, $ f(3) = 1 $

Daniel J.
Numerade Educator

Problem 17

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

$ \displaystyle \lim_{x \to 3^+}f(x) = 4 $, $ \displaystyle \lim_{x \to 3^-}f(x) = 2 $,
$ \displaystyle \lim_{x \to -2}f(x) = 2 $, $ f(3) = 3 $, $ f(-2) = 1 $

Leon D.
Numerade Educator

Problem 18

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

$ \displaystyle \lim_{x \to 0^-}f(x) = 2 $, $ \displaystyle \lim_{x \to 0^+}f(x) = 0 $,
$ \displaystyle \lim_{x \to 4^-}f(x) = 3 $, $ \displaystyle \lim_{x \to 4^+}f(x) = 0 $, $ f(0) = 2 $, $ f(4) = 1 $

IG
Ian G.
Numerade Educator

Problem 19

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{x \to 3}\frac{x^2 - 3x}{x^2 - 9} $,
$ x $ = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999

DM
David M.
Numerade Educator

Problem 20

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{x \to -3}\frac{x^2 - 3x}{x^2 - 9} $,
$ x $ = -2.5, -2.9, -2.95, -2.99, -2.999, -2.9999, -3.5, -3.1, -3.05, -3.01, -3.001, -3.0001

Carolyn B.
Numerade Educator

Problem 21

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{t \to 0}\frac{e^{5t} - 1}{t} $, $ t = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001 $

Paul C.
Numerade Educator

Problem 22

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} $,
$ h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001 $

Oswaldo J.
Numerade Educator

Problem 23

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{x \to 4}\frac{\ln x - \ln 4}{x-4} $

AK
Anjali K.
Numerade Educator

Problem 24

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{p \to -1}\frac{1+p^9}{1+p^{15}} $

Daniel J.
Numerade Educator

Problem 25

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

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

Satyam G.
Numerade Educator

Problem 26

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{t \to 0}\frac{5^t - 1}{t} $

Ma. Theresa A.
Numerade Educator

Problem 27

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

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

Leon D.
Numerade Educator

Problem 28

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{x \to 0^+}x^2 \ln x $

Daniel J.
Numerade Educator

Problem 29

(a) By graphing the function $ f(x) = (\cos 2x - \cos x)/x^2 $ and zooming in toward the point where the graph crosses the $ y $ -axis , estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $.

(b) Check your answer in part (a) by evaluating $ f(x) $ for values of $ x $ that approach 0.

Daniel J.
Numerade Educator

Problem 30

(a) Estimate the value of $$ \lim_{x \to 0}\frac{\sin x}{\sin \pi x} $$
by graphing the function $ f(x) = (\sin x)/(\sin \pi x) $. State your answer correct to two decimal places.

(b) Check your answer in part (a) by evaluating $ f(x) $ for values of $ x $ that approach 0.

Daniel J.
Numerade Educator

Problem 31

Determine the infinite limit.

$ \displaystyle \lim_{x \to 5^+}\frac{x+1}{x-5} $

DM
David M.
Numerade Educator

Problem 32

Determine the infinite limit.

$ \displaystyle \lim_{x \to 5^-}\frac{x+1}{x-5} $

Suman Saurav T.
Numerade Educator

Problem 33

Determine the infinite limit.

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

tj
Tanvi J.
Numerade Educator

Problem 34

Determine the infinite limit.

$ \displaystyle \lim_{x \to 3^-}\frac{\sqrt{x}}{(x-3)^5} $

Ma. Theresa A.
Numerade Educator

Problem 35

Determine the infinite limit.

$ \displaystyle \lim_{x \to 3^+}\ln (x^2 - 9) $

Suman Saurav T.
Numerade Educator

Problem 36

Determine the infinite limit.

$ \displaystyle \lim_{x \to 0^+}\ln (\sin x) $

Ma. Theresa A.
Numerade Educator

Problem 37

Determine the infinite limit.

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

DM
David M.
Numerade Educator

Problem 38

Determine the infinite limit.

$ \displaystyle \lim_{x \to \pi^-}\cot x $

Ma. Theresa A.
Numerade Educator

Problem 39

Determine the infinite limit.

$ \displaystyle \lim_{x \to 2\pi^-}x\csc x $

Ma. Theresa A.
Numerade Educator

Problem 40

Determine the infinite limit.

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

Ma. Theresa A.
Numerade Educator

Problem 41

Determine the infinite limit.

$ \displaystyle \lim_{x \to 2^+}\frac{x^2 - 2x -8}{x^2 -5x +6} $

Leon D.
Numerade Educator

Problem 42

Determine the infinite limit.

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

Ma. Theresa A.
Numerade Educator

Problem 43

Determine the infinite limit.

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

Ma. Theresa A.
Numerade Educator

Problem 44

(a) Find the vertical asymptotes of the function
$$ y = \frac{x^2 +1}{3x - 2x^2} $$

(b) Confirm your answer to part (a) by graphing the function.

Leon D.
Numerade Educator

Problem 45

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

(a) by evaluating $ f(x) = 1/(x^3 - 1) $ for values of $ x $ that approach 1 from the left and from the right,
(b) by reasoning as in Example 9, and
(c) from a graph of $ f $.

Daniel J.
Numerade Educator

Problem 46

(a) By graphing the function $ f(x) = (\tan 4x)/x $ and zooming in toward the point where the graph crosses the $ y $ -axis, estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $.

(b) Check your answer in part (a) by evaluating $ f(x) $ for values of $ x $ that approach 0.

Daniel J.
Numerade Educator

Problem 47

(a) Estimate the value of the limit $ \displaystyle \lim_{x \to 0}(1 + x)^{1/x} $ to five decimal places. Does this number look familiar?

(b) Illustrate part (a) by graphing the function $ y = (1 + x)^{1/x} $.

Daniel J.
Numerade Educator

Problem 48

(a) Graph the function $ f(x) = e^x + \ln | x - 4 | $ for $ 0 \le x \le 5 $. Do you think the graph is an accurate representation of $ f $?

(b) How would you get a graph that represents $ f $ better?

Daniel J.
Numerade Educator

Problem 49

(a) Evaluate the function $ f(x) = x^2 - (2^x/1000) $ for $ x $ = 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of $$ \lim_{x \to 0}\left( x^2 - \frac{2^x}{1000} \right) $$

(b) Evaluate $ f(x) $ for $ x $ = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

Daniel J.
Numerade Educator

Problem 50

(a) Evaluate $ h(x) = (\tan x - x)/x^3 $ for $ x $ = 1, 0.5, 0.1, 0.05, 0.01, and 0.005.

(b) Guess the value of $ \displaystyle \lim_{x \to 0}\frac{\tan x - x}{x^3} $.

(c) Evaluate $ h(x) $ for successively smaller values of $ x $ until you finally reach a value of 0 for $ h(x) $. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method of evaluating this limit will be explained.)

(d) Graph the function $ h $ in the viewing rectangle $ [-1, 1] $ by $ [0, 1] $. Then zoom in toward the point where the graph crosses the $ y $ -axis to estimate the limit of $ h(x) $ as $ x $ approaches 0. Continue to zoom in until you observe distortions in the graph of $ h $. Compare with the results of part (c).

Oswaldo J.
Numerade Educator

Problem 51

Graph the function $ f(x) = \sin (\pi/x) $ of Example 4 in the viewing rectangle $ [-1, 1] $ by $ [-1, 1] $. Then zoom in toward the origin several times. Comment on the behavior of this function.

Leon D.
Numerade Educator

Problem 52

Consider the function $ f(x) = \tan \frac{1}{x} $.

(a) Show that $ f(x) = 0 $ for $ x = \frac{1}{\pi}, \frac{1}{2\pi}, \frac{1}{3\pi}, ... $

(b) Show that $ f(x) = 1 $ for $ x = \frac{4}{\pi}, \frac{4}{5\pi}, \frac{4}{9\pi}, ... $

(c) What can you conclude about $ \displaystyle \lim_{x \to 0^+}\tan \frac{1}{x} $?

Daniel J.
Numerade Educator

Problem 53

Use a graph to estimate the equations of all the vertical asymptotes of the curve
$ y = \tan (2 \sin x) $ $ -\pi \le x \le \pi $
Then find the exact equations of these asymptotes.

Daniel J.
Numerade Educator

Problem 54

In the theory of relativity, the mass of a particle with velocity $ v $ is
$$ m = \frac{m_0}{\sqrt{1 - v^2/c^2}} $$
where $ m_0 $ is the mass of the particle at rest and $ c $ is the speed of light. What happens as $ v \to c^- $?

Linda H.
Numerade Educator

Problem 55

(a) Use numerical and graphical evidence to guess the value of the limit
$$ \lim_{x \to 1}\frac{x^3 - 1}{\sqrt{x} - 1} $$

(b) How close to 1 does $ x $ have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?

DM
David M.
Numerade Educator