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Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275361
BuyFindarrow_forward

Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275361
Chapter 1.5, Problem 70E
Textbook Problem
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Proof Prove the difference, product, and quotient properties in Theorem 1.15.

To determine

To prove: The properties of infinite limits.

Difference: limxc[f(x)g(x)]=.

Product: limxc[f(x)g(x)]=,L>0 and limxc[f(x)g(x)]=,L<0.

Quotient: limxc[g(x)f(x)]=0.

Explanation of Solution

Given: The provided functions are f(x) and g(x). And limxcf(x)=, limxcg(x)=L where c and L are real numbers.

Proof:

Let f(x) be a function that is defined at every real number in some open interval containing c (except possibly at c itself) and limxcf(x)=

Apply the definition of infinite limits:

For each M1>0 there exists a δ1>0 such that f(x)>M1 whenever

0<|xc|<δ1.

And since limxcg(x)=L. For simplicity, assume L>0.

So, there exists δ2>0 such that |g(x)L|<1 whenever 0<|xc|<δ2.

Now, first show that the limit of [f(x)g(x)] is infinite, choose M>0. So, find δ>0 such that [f(x)g(x)]>M whenever 0<|xc|<δ.

Let M1=M+L1. By letting δ=min{δ1,δ2}, it implies that 0 <∣x − c∣< δ implies f(x)>M+L1 and |g(x)L|<1. So, 1<g(x)L<1

Rewrite it as:

1+L<g(x)<1+L.

So, g(x)>L1

Now, subtract both the inequality, you can write

[f(x)g(x)]>(M+L1)(L1)=M.

Therefore, [f(x)g(x)]>M

Hence, limxc[f(x)g(x)]=

Now, show that the limit of product of functions:

Case I: limxc[f(x)g(x)]= when L>0

Choose M > 0. So, find δ>0 such that [f(x)g(x)]>M whenever 0<|xc|<δ.

Let M1=2ML. And since limxcg(x)=L

So, there exists δ2>0 such that |g(x)L|<L2 whenever 0<|xc|<δ2.

By letting δ=min{δ1,δ2}, it implies that 0 <∣x − c∣< δ implies f(x)>2ML and |g(x)L|<L2. So, L2<g(x)L<L2

Rewrite it as:

L2+L<g(x)<L2+L.

So, L2<g(x)<3L2

Implies, g(x)>L2

Now, product of both the inequality gives:

[f(x)g(x)]>(2ML)×L2=M.

Therefore, [f(x)g(x)]>M

Hence, limxc[f(x)g(x)]= when L>0

Case II: limxc[f(x)g(x)]= when L<0

Choose M > 0. So, find δ>0 such that [f(x)g(x)]>M whenever 0<|xc|<δ.

Let M1=2ML. And since limxcg(x)=L

So, there exists δ2>0 such that |g(x)L|<L2 whenever 0<|xc|<δ2.

By letting δ=min{δ1,δ2}, it implies that 0 <∣x − c∣< δ implies f(x)>2ML and |g(x)L|<L2. So, L2<g(x)L<L2

Rewrite it as:

L2+L<g(x)<L2+L.

So, 3L2<g(x)<L2.

Since L<0. So, L2>0

Take minus of the above inequality:

L2<g(x)<3L2

Now, product of both the inequality gives

[f(x)g(x)]=f(x)[g(x)]>(2ML)×L2=M

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Chapter 1 Solutions

Calculus of a Single Variable
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True or False? 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Every day you dissolve 28...Ch. 1.4 - Data Plan A cell phone service charges $10 for the...Ch. 1.4 - Inventory Management The number of units in...Ch. 1.4 - Dj Vu At 8:00 a.m. on Saturday, a nun begins...Ch. 1.4 - Volume Use the Intermediate Value Theorem to show...Ch. 1.4 - Proof Prove that if f is continuous and has no...Ch. 1.4 - Dirichlet Function Show that the Dirichlet...Ch. 1.4 - Continuity of a Function Show that the function...Ch. 1.4 - Signum Function The signum function is defined by...Ch. 1.4 - Modeling Data The table lists the frequency F (in...Ch. 1.4 - Creating Models A swimmer crosses a pool of width...Ch. 1.4 - Making a Function Continuous Find all values of c...Ch. 1.4 - Proof Prove that for any real number y there...Ch. 1.4 - Making a Function Continuous Let f(x)=x+c2cx,c0...Ch. 1.4 - Proof Prove that if limx0f(c+x)=f(c) then f is...Ch. 1.4 - Continuity of a Function Discuss the continuity of...Ch. 1.4 - Proof (a) Let f1(x) and f2(x) be continuous on the...Ch. 1.4 - Prove or disprove: If x and y are real numbers...Ch. 1.4 - Determine all polynomials P(x) such that...Ch. 1.5 - Infinite Limit In your own words, describe the...Ch. 1.5 - Vertical Asymptote In your own words, describe...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Vertical Asymptote or Removable Discontinuity In...Ch. 1.5 - Vertical Asymptote or Removable Discontinuity In...Ch. 1.5 - Vertical Asymptote or Removable Discontinuity In...Ch. 1.5 - Vertical Asymptote or Removable Discontinuity In...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Finding a One-Sided Limit Using Technology In...Ch. 1.5 - Finding a One-Sided Limit Using Technology In...Ch. 1.5 - Determining Limits In Exercises 53 and 54, use the...Ch. 1.5 - Determining Limits In Exercises 53 and 54, use the...Ch. 1.5 - EXPLORING CONCEPTS Writing a Rational Function...Ch. 1.5 - EXPLORING CONCEPTS Rational Function Does the...Ch. 1.5 - Sketching a Graph Use the graph of the function f...Ch. 1.5 - Relativity According to the theory of relativity,...Ch. 1.5 - Numerical and Graphical Reasoning Use a graphing...Ch. 1.5 - HOW DO YOU SEE IT? 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Choosing Graphs Consider the graphs of the four...Ch. 1 - Limits and Continuity Sketch the graph of the...Ch. 1 - Limits and Continuity Sketch the graph of the...Ch. 1 - Escape Velocity To escape Earth's gravitational...Ch. 1 - Pulse Function For positive numbers ab, the pulse...Ch. 1 - Proof Let a be a nonzero constant. 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