Q. 6 Find functions f and g and a rea... [FREE SOLUTION]

Chapter 1: Q. 6 (page 135)

Find functions f 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 sum rule for limits? Why or why not?

Short Answer

Expert verified

The functions $f (x) = \frac{1}{x}$ and $g (x) = - \frac{1}{x}$ and a real number $c = 0$ such that $\lim_{x 鈫� c} f (x) + \lim_{x 鈫� c} g (x) 鈮� \lim_{x 鈫� c} (f (x) + g (x))$ andcontradict the sum rule for limits.

Step by step solution

Step 1. Given information.

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

We have

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

and

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

So $\lim_{x 鈫� c} f (x) + \lim_{x 鈫� c} g (x) 鈮� \lim_{x 鈫� c} (f (x) + g (x))$ and given example contradict the sum rule for limits, which states $\lim_{x 鈫� c} f (x) + \lim_{x 鈫� c} g (x) = \lim_{x 鈫� c} (f (x) + g (x))$

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91影视

Find functions f 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 sum 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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Most popular questions from this chapter

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91影视

Find functions f 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 sum 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

Geometry

Statistics

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find functions f 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 sum rule for limits? Why or why not?