Chapter 14: Q. 48 (page 889)

In Problems 45-60, determine whether $f$ is continuous at $c$ .
localid="1647002262582" $f (x) = \frac{x^{3} - 8}{x^{2} + 4} c = 2$

Short Answer

Expert verified

The function is continuous at $c$

Step by step solution

Given information

Given function

$f (x) = \frac{x^{3} - 8}{x^{2} + 4} c = 2$

Check whether the function is defined at c

We have

$f (2) = \frac{2^{3} - 8}{2^{2} + 4} f (2) = 0$

So it is defined at $c$

Check the left hand limits and right hand limits

Checking, we get

$\lim_{x \to 2^{-}} f (x) = \lim_{x \to 2^{-}} \frac{2^{3} - 8}{2^{2} + 4} \lim_{x \to 2^{-}} f (x) = \lim_{x \to 2^{-}} \frac{2^{3} - 8}{2^{2} + 4} \lim_{x \to 2^{-}} f (x) = 0$

Again,

$\lim_{x \to 2^{+}} f (x) = \lim_{x \to 2^{+}} \frac{2^{3} - 8}{- 2^{2} + 4} \lim_{x \to 2^{+}} f (x) = 0$

Since $\lim_{x \to c} f (x) = f (c)$ so it is continuous at $c$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 45-60, determine whether $f$ is continuous at $c$ .
localid="1647002262582" $f (x) = \frac{x^{3} - 8}{x^{2} + 4} c = 2$

Short Answer

Step by step solution

Given information

Check whether the function is defined at c

Check the left hand limits and right hand limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Discrete Mathematics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 45-60, determine whether fis continuous at c.localid="1647002262582" f(x)=x3-8x2+4c=2

Short Answer

Step by step solution

Given information

Check whether the function is defined at c

Check the left hand limits and right hand limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Discrete Mathematics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 45-60, determine whether $f$ is continuous at $c$ .
localid="1647002262582" $f (x) = \frac{x^{3} - 8}{x^{2} + 4} c = 2$