Q. 7 Find functions聽f and g and a re... [FREE SOLUTION]

Chapter 1: Q. 7 (page 135)

Find functionsf and g and a real number c such that $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$ . Does this example contradict the product rule for limits? Why or why not?

Short Answer

Expert verified

The function $f (x) = x$ and $g (x) = \frac{1}{x}$ and a real number $c = 0$ such that $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$ and contradicts product rule for limits.

Step by step solution

Step 1. Given information.

An example is to be written for the condition $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$

Step 2. Example for the given condition.

Let $f (x) = x$ and $g (x) = \frac{1}{x}$ and a real number $c = 0$ such that limit is $x 鈫� 0$ .

We have

$(\lim_{x 鈫� 0} f (x)) (\lim_{x 鈫� 0} g (x)) = (\lim_{x 鈫� 0} x) (\lim_{x 鈫� 0} \frac{1}{x}) = 0 (\frac{1}{0}) = undefined$

and

$\lim_{x 鈫� 0} (f (x) g (x)) = \lim_{x 鈫� 0} (x 脳 \frac{1}{x}) = \lim_{x 鈫� 0} (1) = 1$

So, $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$ and this example contradict the product rule for limits which states that $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) = \lim_{x 鈫� c} (f (x) g (x))$ .

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91影视

Find functionsf and g and a real number c such that $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$ . Does this example contradict the product rule for limits? Why or why not?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Example for the given condition.

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91影视

Find functionsf and g and a real number c such that limx鈫�cf(x)limx鈫�cg(x)鈮�limx鈫�c(f(x)g(x)). Does this example contradict the product rule for limits? Why or why not?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Example for the given condition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Mechanics Maths

Geometry

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find functionsf and g and a real number c such that $(\lim_{x 鈫� c} f (x)) (\lim_{x 鈫� c} g (x)) 鈮� \lim_{x 鈫� c} (f (x) g (x))$ . Does this example contradict the product rule for limits? Why or why not?