Chapter 1: Q. 39 (page 120)

In Exercises 39–44, 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.
$f (x) = \{\begin{cases} (x - 3), i f x < 3 \\ - (x - 3), i f x ⩾ 3 \end{cases} .$

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$T h e g i v e n f u n c t i o n i s c o n t i n u o u s a t a l l p o i n t s i n i t s d o m a i n a n d i t i s b o t h l e f t a n d r i g h t c o n t i n u o u s a t x = 3 .$

Step by step solution

Step 1. Given Information.

Given the function:

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

Step 2. Finding the limits at the break point.

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

Step 3. Finding the type of discontinuity.

Since the function is not discontinuous anywhere so no question arises for the type of discontinuity and as it is continuous all over in its domain therefore, it is both left continuous and right continuous at x=3.

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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.

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In Exercises 39–44, 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<3-(x-3),ifx⩾3.

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

Statistics

Pure Maths

Applied Mathematics

Discrete Mathematics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.