Q. 40 In Exercises 39鈥�44, use Theore... [FREE SOLUTION]

Chapter 1: Q. 40 (page 120)

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.
$f (x) = \{\begin{array}{l} (x - 3), i f x < 0 \\ - (x - 3), i f x 猢� 0 \end{array} .$

Short Answer

Expert verified

$T h e g i v e n f u n c t i o n i s n o t c o n t i n u o u s a t x = 0, i t h a s j u m p d i s c o n t i n u i t y a n d i t i s r i g h t c o n t i n u o u s .$

Step by step solution

Step 1. Given Information.

Given the function;

$f (x) = \{\begin{array}{l} (x - 3), i f x < 0 \\ - (x - 3), i f x 猢� 0 \end{array} a n d i t h a s i t' s b r e a k p o i n t a t x = 0 .$

Step 2. Finding the limits at the break point.

$A t x = 0, L H L = \lim_{x 鈫� 0^{-}} f (x) = \lim_{x 鈫� 0^{-}} (x - 3) = 0 - 3 = - 3 . R H L = \lim_{x 鈫� 0^{+}} f (x) = \lim_{x 鈫� 0^{+}} - (x - 3) = - (0 - 3) = 3 . f (0) = - (x - 3) = - (0 - 3) = 3 . S o, R H L = f (0) 鈮� L H L .$

Step 3. Finding the type of discontinuity.

Since the function is discontinuous only at $x = 0,$ from Step 2.

Now we know $L H L 鈮� R H L$ this means both left and righthand limit exists but they are not equal so this is jump discontinuity.

And also, RHL = f(0) this means this function is right continuous.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break point.

Step 3. Finding the type of discontinuity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.f(x)=(x-3),ifx<0-(x-3),ifx猢�0.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break point.

Step 3. Finding the type of discontinuity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.