Calculus

James Stewart

Chapter 2

Limits and Derivatives - all with Video Answers

Educators

AD

+ 6 more educators

Section 3

Calculating Limits Using the Limit Laws

Problem 1

Given that
$ \displaystyle \lim_{x \to 2}f(x) = 4 $ $ \displaystyle \lim_{x \to 2}g(x) = -2 $ $ \displaystyle \lim_{x \to 2}h(x) = 0 $
find the limits that exist. If the limit does not exist, explain why.

(a) $ \displaystyle \lim_{x \to 2}[f(x) + 5g(x)] $
(b) $ \displaystyle \lim_{x \to 2}[g(x)]^3 $
(c) $ \displaystyle \lim_{x \to 2}\sqrt{f(x)} $
(d) $ \displaystyle \lim_{x \to 2}\frac{3f(x)}{g(x)} $
(e) $ \displaystyle \lim_{x \to 2}\frac{g(x)}{h(x)} $
(f) $ \displaystyle \lim_{x \to 2}\frac{g(x)h(x)}{f(x)} $

Aparna S.

Aparna S.

Numerade Educator

Problem 2

The graphs of $ f $ and $ g $ are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.
(a) $ \displaystyle \lim_{x \to 2}[f(x) + g(x)] $
(b) $ \displaystyle \lim_{x \to 0}[f(x) - g(x)] $
(c) $ \displaystyle \lim_{x \to -1}[f(x) g(x)] $
(d) $ \displaystyle \lim_{x \to 3}\frac{f(x)}{g(x)} $
(e) $ \displaystyle \lim_{x \to 2}[x^2 f(x)] $
(f) $ \displaystyle f(-1) + \lim_{x \to -1}g(x) $

DM

David M.

Numerade Educator

Problem 3

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{x \to 3}(5x^3 - 3x^2 + x - 6) $

Suman Saurav T.

Suman Saurav T.

Numerade Educator

Problem 4

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{x \to -1}(x^4 - 3x)(x^2 + 5x + 3) $

Suman Saurav T.

Suman Saurav T.

Numerade Educator

Problem 5

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} $

DM

David M.

Numerade Educator

Problem 6

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{u \to -2}\sqrt{u^4 + 3u + 6} $

DM

David M.

Numerade Educator

Problem 7

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{x \to 8}(1 + \sqrt[3]{x})(2 - 6x^2 + x^3) $

Ma. Theresa A.

Numerade Educator

Problem 8

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

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

Ma. Theresa A.

Numerade Educator

Problem 9

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

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

Ma. Theresa A.

Numerade Educator

Problem 10

(a) What is wrong with the following equation?
$$ \frac {x^2 + x - 6}{x - 2} = x + 3 $$

(b) In view of part (a), explain why the equation
$$ \lim_{x \to 2}\frac {x^2 + x - 6}{x - 2} = \lim_{x \to 2}(x + 3) $$
is correct.

Leon D.

Numerade Educator

Problem 11

Evaluate the limit, if it exists.

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

Anthony H.

Numerade Educator

Problem 12

Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to -3}\frac{x^2 + 3x}{x^2 - x - 12} $

Leon D.

Numerade Educator

Problem 13

Evaluate the limit, if it exists.

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

DM

David M.

Numerade Educator

Problem 14

Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to 4}\frac{x^2 + 3x}{x^2 - x - 12} $

Leon D.

Numerade Educator

Problem 15

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to -3}\frac{t^2 - 9}{2t^2 + 7t + 3} $

Anthony H.

Numerade Educator

Problem 16

Evaluate the limit, if it exists.

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

Ma. Theresa A.

Numerade Educator

Problem 17

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(-5 + h)^2 - 25}{h} $

Leon D.

Numerade Educator

Problem 18

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(2 + h)^3 - 8}{h} $

Leon D.

Numerade Educator

Problem 19

Evaluate the limit, if it exists.

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

Ma. Theresa A.

Numerade Educator

Problem 20

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 1}\frac{t^4 - 1}{t^3 - 1} $

Ma. Theresa A.

Numerade Educator

Problem 21

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{\sqrt{9 + h}-3}{h} $

DM

David M.

Numerade Educator

Problem 22

Evaluate the limit, if it exists.

$ \displaystyle \lim_{u \to 2}\frac{\sqrt{4u + 1}-3}{u - 2} $

Ma. Theresa A.

Numerade Educator

Problem 23

Evaluate the limit, if it exists.

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

Anthony H.

Numerade Educator

Problem 24

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(3 + h)^{-1} - 3^{-1}}{h} $

Ma. Theresa A.

Numerade Educator

Problem 25

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 0}\frac{\sqrt{1 + t} - \sqrt{1 - t}}{t} $

KC

Kyle C.

Numerade Educator

Problem 26

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 0}\left( \frac{1}{t} - \frac{1}{t^2 + t} \right) $

Ma. Theresa A.

Numerade Educator

Problem 27

Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to 16}\frac{4 - \sqrt{x}}{16x - x^2} $

Ma. Theresa A.

Numerade Educator

Problem 28

Evaluate the limit, if it exists.

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

Leon D.

Numerade Educator

Problem 29

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 0}\left( \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \right)$

Ma. Theresa A.

Numerade Educator

Problem 30

Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to -4}\frac{\sqrt{x^2 + 9} - 5}{x + 4} $

Daniel J.

Numerade Educator

Problem 31

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(x + h)^3 - x^3}{h} $

Anthony H.

Numerade Educator

Problem 32

Evaluate the limit, if it exists.
$ \displaystyle \lim_{h \to 0}\frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h} $

Ma. Theresa A.

Numerade Educator

Problem 33

(a) Estimate the value of
$$ \lim_{x \to 0}\frac{x}{\sqrt{1 + 3x} - 1} $$
by graphing the function $ f(x) = x/(\sqrt{1 + 3x} - 1) $.

(b) Make a table of values of $ f(x) $ for $ x $ close to 0 and guess the value of the limit.

(c) Use the Limit Laws to prove that your guess is correct.

Daniel J.

Numerade Educator

Problem 34

(a) Use a graph of
$$ f(x) = \frac{\sqrt{3 + x} - \sqrt{3}}{x} $$
to estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $ to two decimal places.

(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

(c) Use the Limit Laws to find the exact value of the limit.

Daniel J.

Numerade Educator

Problem 35

Use the Squeeze Theorem to show that $ \displaystyle \lim_{x \to 0}(x^2 \cos 20\pi x) = 0 $. Illustrate by graphing the functions $ f(x) = -x^2 $, $ g(x) = x^2 \cos 20\pi x $, and $ h(x) = x^2 $ on the same screen.

Daniel J.

Numerade Educator

Problem 36

Use the Squeeze Theorem to show that
$$ \lim_{x \to 0}\sqrt{x^3 + x^2}\sin \frac{\pi}{x} = 0 $$

Illustrate by graphing the functions $ f $, $ g $, and $ h $ (in the notation of the Squeeze Theorem) on the same screen.

Daniel J.

Numerade Educator

Problem 37

If $ 4x - 9 \le f(x) \le x^2 - 4x + 7 $ for $ x \ge 0 $, find $ \displaystyle \lim_{x \to 4}f(x) $.

Daniel J.

Numerade Educator

Problem 38

If $ 2x \le g(x) \le x^4 - x^2 +2 $ for all $ x $, evaluate $ \displaystyle \lim_{x \to 1}g(x) $.

Ma. Theresa A.

Numerade Educator

Problem 39

Prove that $ \displaystyle \lim_{x \to 0}x^4 \cos \frac{2}{x} = 0 $.

Suman Saurav T.

Suman Saurav T.

Numerade Educator

Problem 40

Prove that $ \displaystyle \lim_{x \to 0^+}\sqrt{x} e^{\sin (\pi/x)} = 0 $.

Daniel J.

Numerade Educator

Problem 41

Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to 3}\left( 2x + | x - 3 | \right) $

Ma. Theresa A.

Numerade Educator

Problem 42

Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to -6}\frac{2x + 12}{| x + 6 |} $

Leon D.

Numerade Educator

Problem 43

Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to 0.5^-}\frac{2x - 1}{| 2x^3 - x^2 |} $

KS

Karthika S.

Numerade Educator

Problem 44

Find the limit, if it exists. If the limit does not exist, explain why.

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

Daniel J.

Numerade Educator

Problem 45

Find the limit, if it exists. If the limit does not exist, explain why.

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

Ma. Theresa A.

Numerade Educator

Problem 46

Find the limit, if it exists. If the limit does not exist, explain why.

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

Daniel J.

Numerade Educator

Problem 47

The $ signum $ (or sign) $ function$ , denoted by sgn, is defined by
sgn $ x = \left\{
\begin{array}{ll}
-1 & \mbox{if $ x < 0 $}\\
0 & \mbox{if $ x = 0 $}\\
1 & \mbox{if $ x > 0 $}
\end{array} \right.$

(a) Sketch the graph of this function.
(b) Find each of the following limits or explain why it does not exist.
(i) $ \displaystyle \lim_{x \to 0^+}\text{sgn $ x $} $
(ii) $ \displaystyle \lim_{x \to 0^-}\text{sgn $ x $} $
(iii) $ \displaystyle \lim_{x \to 0}\text{sgn $ x $} $
(iv) $ \displaystyle \lim_{x \to 0}| \text{sgn $ x $} | $

Leon D.

Numerade Educator

Problem 48

Let $ g(x) = \text{sgn}(\sin x) $.

(a) Find each of the following limits or explain why it does not exist.
(i) $ \displaystyle \lim_{x \to 0^+}g(x) $
(ii) $ \displaystyle \lim_{x \to 0^-}g(x) $
(iii) $ \displaystyle \lim_{x \to 0}g(x) $
(iv) $ \displaystyle \lim_{x \to \pi^+}g(x) $
(v) $ \displaystyle \lim_{x \to \pi^-}g(x) $
(vi) $ \displaystyle \lim_{x \to \pi}g(x) $

(b) For which values of $ a $ does $ \displaystyle \lim_{x \to a}g(x) $ not exist?
(c) Sketch a graph of $ g $.

Oswaldo J.

Numerade Educator

Problem 49

Let $ g(x) = \frac{x^2 + x - 6}{| x - 2 |} $.

(a) Find
(i) $ \displaystyle \lim_{x \to 2^+}g(x) $ (ii) $ \displaystyle \lim_{x \to 2^-}g(x) $
(b) Does $ \displaystyle \lim_{x \to 2}g(x) $ exist?
(c) Sketch the graph of $ g $.

Ma. Theresa A.

Numerade Educator

Problem 50

Let
$ f(x) = \left\{
\begin{array}{ll}
x^2 + 1 & \mbox{if $ x < 1 $}\\
(x - 2)^2 & \mbox{if $ x \ge 1 $}
\end{array} \right.$

(a) Find $ \displaystyle \lim_{x \to 1^-}f(x) $ and $ \displaystyle \lim_{x \to 1^+}f(x) $.
(b) Does $ \displaystyle \lim_{x \to 1}f(x) $ exist?
(c) Sketch the graph of $ f $.

Daniel J.

Numerade Educator

Problem 51

Let
$ B(t) = \left\{
\begin{array}{ll}
4 - \frac{1}{2}t & \mbox{if $ t < 2 $}\\
\sqrt{t + c} & \mbox{if $ t \ge 2 $}
\end{array} \right.$
Find the value of $ c $ so that $ \displaystyle \lim_{t \to 2}B(t) $ exists.

Daniel J.

Numerade Educator

Problem 52

Let
$ g(x) = \left\{
\begin{array}{ll}
x & \mbox{if $ x < 1 $}\\
3 & \mbox{if $ x = 1 $}\\
2 - x^2 & \mbox{if $ 1 < x \le 2 $}\\
x - 3 & \mbox{if $ x > 2 $}
\end{array} \right.$

(a) Evaluate each of the following, if it exists.

(i) $ \displaystyle \lim_{x \to 1^-}g(x) $
(ii) $ \displaystyle \lim_{x \to 1}g(x) $
(iii) $ g(1) $
(iv) $ \displaystyle \lim_{x \to 2^-}g(x) $
(v) $ \displaystyle \lim_{x \to 2^+}g(x) $
(vi) $ \displaystyle \lim_{x \to 2}g(x) $

(b) Sketch the graph of $ g $.

Daniel J.

Numerade Educator

Problem 53

(a) If the symbol [] I denotes the greatest integer function defined in Example 10 , evaluate
(i) $\lim _{x \rightarrow-2^{+}}[x]$
(ii) $\lim _{x \rightarrow-2}[x]$
(iii) $\lim _{x \rightarrow-24}[x]$
(b) If $n$ is an inleger, evaluale
(i) $\lim _{x \rightarrow \infty}[x]$
(iii) $\lim _{x \rightarrow \infty}[x]$
(c) For what values of $a$ dacs lim. $. . .[\mathrm{xl}]$ caist?

Daniel J.

Numerade Educator

Problem 54

Let $ f(x) = [ \cos x ] $, $ -\pi \le x \le \pi $.
(a) Sketch the graph of $ f $.
(b) Evaluate each limit, if it exists.
(i) $ \displaystyle \lim_{x \to 0}f(x) $
(ii) $ \displaystyle \lim_{x \to (\pi/2)^-}f(x) $
(iii) $ \displaystyle \lim_{x \to (\pi/2)^+}f(x) $
(iv) $ \displaystyle \lim_{x \to \pi/2}f(x) $
(c) For what values of $ a $ does $ \displaystyle \lim_{x \to a}f(x) $ exist?

Daniel J.

Numerade Educator

Problem 55

If $ f(x) = [ x ] + [ -x ] $, show that $ \displaystyle \lim_{x \to 2}f(x) $ exists but is not equal to $f(2) $.

Daniel J.

Numerade Educator

Problem 56

In the theory of relativity, the Lorentz contraction formula
$$ L = L_0 \sqrt{1 - v^2/c^2} $$
expresses the length $ L $ of an object as a function of its velocity $ v $ with respect to an observer, where $ L_0 $ is the length of the object at rest and $ c $ is the speed of light. Find $ \displaystyle \lim_{v \to c}-L $ and interpret the result. Why is a left-hand limit necessary?

Daniel J.

Numerade Educator

Problem 57

If $ p $ is a polynomial, show that $ \displaystyle \lim_{x \to a}p(x) = p(a) $.

Daniel J.

Numerade Educator

Problem 58

If $ r $ is a rational function, use Exercise 57 to show that $ \displaystyle \lim_{x \to a}r(x) = r(a) $ for every number $ a $ in the domain of $ r $.

Daniel J.

Numerade Educator

Problem 59

If $ \displaystyle \lim_{x \to 1}\frac{f(x) - 8}{x - 1} = 10 $, find $ \displaystyle \lim_{x \to 1}f(x) $.

Daniel J.

Numerade Educator

Problem 60

If $ \displaystyle \lim_{x \to 0}\frac{f(x)}{x^2} = 5 $, find the following limits.

(a) $ \displaystyle \lim_{x \to 0}f(x) $ (b) $ \displaystyle \lim_{x \to 0}\frac{f(x)}{x} $

Daniel J.

Numerade Educator

Problem 61

If
$ f(x) = \left\{
\begin{array}{ll}
x^2 & \mbox{if $ x $ is rational}\\
0 & \mbox{if $ x $ is irrational}
\end{array} \right.$
prove that $ \displaystyle \lim_{x \to 0}f(x) = 0 $.

Daniel J.

Numerade Educator

Problem 62

Show by means of an example that $ \displaystyle \lim_{x \to a}\left[ f(x) + g(x) \right] $ may exist even though neither $ \displaystyle \lim_{x \to a}f(x) $ nor $ \displaystyle \lim_{x \to a}g(x) $ exists.

Daniel J.

Numerade Educator

Problem 63

Show by means of an example that $ \displaystyle \lim_{x \to a}\left[ f(x) g(x) \right] $ may exist even though neither $ \displaystyle \lim_{x \to a}f(x) $ nor $ \displaystyle \lim_{x \to a}g(x) $ exists.

Daniel J.

Numerade Educator

Problem 64

Evaluate $ \displaystyle \lim_{x \to 2}\frac{\sqrt{6 - x} - 2}{\sqrt{3 - x} - 1} $.

Daniel J.

Numerade Educator

Problem 65

Is there a number $ a $ such that
$$ \lim_{x \to -2}\frac{3x^2 + ax + a + 3}{x^2 + x - 2} $$
exists? If so, find the value of $ a $ and the value of the limit.

Daniel J.

Numerade Educator

Problem 66

The figure shows a fixed circle $ C_1 $ with equation $ (x - 1)^2 + y^2 = 1 $ and a shrinking circle $ C_2 $ with radius $ r $ and center the origin. $ P $ is the point $ (0,r) $, $ Q $ is the upper point of intersection of the two circles, and $ R $ is the point of intersection of the line $ PQ $ and the $ x $-axis. What happens to $ R $ as $ C_2 $ shrinks, that is, as $ r \to 0^+ $?

Colin O.

Numerade Educator