Q. 43 In Exercises 39鈥�44, use Theore... [FREE SOLUTION]

Chapter 1: Q. 43 (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.
$f (x) = \{\begin{matrix} x + 1, i f x < 1 \\ 3 x - 1, i f 1 猢� x < 2 \\ x + 2, i f x 猢� 2 \end{matrix} .$

Short Answer

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$T h e f u n c t i o n i s d i s c o n t i n u o u s a t x = 2 o n l y a n d t h i s i s j u m p d i s c o n t i n u i t y a n d t h e f u n c t i o n i s r i g h t c o n t i n u o u s a t x = 2 .$

Step by step solution

Step 1. Given Information.

Given the function:

$f (x) = \{\begin{matrix} x + 1, i f x < 1 \\ 3 x - 1, i f 1 猢� x < 2 \\ x + 2, i f x 猢� 2 \end{matrix} a n d i t s b r e a k p o i n t s w h i c h a r e x = 1, 2 .$

Step 2. Finding the limits at the break points.

$A t x = 1, L H L = \lim_{x 鈫� 1^{-}} f (x) = \lim_{x 鈫� 1^{-}} x + 1 = 1 + 1 = 2 . R H L = \lim_{x 鈫� 1^{+}} f (x) = \lim_{x 鈫� 1^{+}} 3 x - 1 = 3 - 1 = 2 . f (1) = 3 (1) - 1 = 3 - 1 = 2 . S i n c e, L H L = R H L = f (1) . T h i s m e a n s i t i s c o n t i n u o u s a t x = 1 .$

$N o w a t x = 2, L H L = \lim_{x 鈫� 2^{-}} f (x) = \lim_{x 鈫� 2^{-}} 3 x - 1 = 6 - 1 = 5 . R H L = \lim_{x 鈫� 2^{+}} f (x) = \lim_{x 鈫� 2^{+}} x + 2 = 2 + 2 = 4 . f (2) = x + 2 = 2 + 2 = 4 . S o, R H L = f (2) 鈮� L H L .$

Step 3. Finding the type of discontinuity.

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

Now we know $R H L 鈮� L H L a n d R H L = f (2)$ this means both left and righthand limit exists but they are not equal so this is jump discontinuity.

And also, RHL = f(2) this means this function is right continuous.

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break points.

Step 3. Finding the type of discontinuity.

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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+1,ifx<13x-1,if1猢�x<2x+2,ifx猢�2.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break points.

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

Geometry

Decision Maths

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.