Chapter 14: Q. 12 (page 907)

Determine whether $f$ is continuous at each of the following numbers. If it is not, explain why not.
$(a) x = - 2 (b) x = 1 (c) x = 3 (d) x = 4$

Short Answer

Expert verified

The function is continuous at $x = - 2, 4$ and discontinuous at $x = 1, 3$

Step by step solution

Part (a) Step 1. Given Information

A graph is given.

We need to find if the function is continuous at $x = - 2$

Part(a) Step 2. Explanation

For $x = 2, f (2) = 2$ It is clear from the graph that,

$\lim_{x \to - 2^{-}} f (x) = 2 \underset{x \to - 2^{+}}{l i m} f (x) = 2 \lim_{x \to - 2^{-}} f (x) = 2 = \lim_{x \to - 2^{+}} f (x) \lim_{x \to - 2} f (x) = 2$

Therefore, continuous limit exists.

f(1)=0Part(b) Step 1. Explanation

For $x = 1$

It is clear from the graph that,

$\lim_{x \to 1^{-}} f (x) = 2 \lim_{x \to 1^{+}} f (x) = 2 \lim_{x \to 1^{-}} f (x) = 2 = \lim_{x \to 1^{+}} f (x) \lim_{x \to 1} f (x) = 2 B u t f (1) = 0$

Therefore, continuous limit does not exist.

Part(c) Step 1. Explanation

For $x = 3, f (3) = - 3$

It is clear from the graph,

$\lim_{x \to 3^{-}} f (x) = 5 \lim_{x \to 3^{+}} f (x) = - 3$

Since,

$\lim_{x \to 3^{-}} f (x) = 5 \neq - 3 = \lim_{x \to 3^{+}} f (x)$

Therefore, continuous limit does not exist.

Part(d) Step 1. Explanation

For $x = 4, f (4) = - 4$

From the graph,

$\lim_{x \to 4} f (x) = - 4$

Therefore, continuous limit exists.

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.

Determine whether $f$ is continuous at each of the following numbers. If it is not, explain why not.
$(a) x = - 2 (b) x = 1 (c) x = 3 (d) x = 4$

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part(a) Step 2. Explanation

f(1)=0Part(b) Step 1. Explanation

Part(c) Step 1. Explanation

Part(d) Step 1. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Determine whether fis continuous at each of the following numbers. If it is not, explain why not.(a)x=-2(b)x=1(c)x=3(d)x=4

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part(a) Step 2. Explanation

f(1)=0Part(b) Step 1. Explanation

Part(c) Step 1. Explanation

Part(d) Step 1. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Determine whether $f$ is continuous at each of the following numbers. If it is not, explain why not.
$(a) x = - 2 (b) x = 1 (c) x = 3 (d) x = 4$