Q. 42 In Exercises 39鈥�44, use Theore... [FREE SOLUTION]

Chapter 1: Q. 42 (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.
$f (x) = \{\begin{matrix} \sqrt{- x}, i f x < 0 \\ 2, i f x = 0 \\ \sqrt{x}, i f x > 0 \end{matrix} .$

Short Answer

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

Step by step solution

Step 1. Given Information.

Given the function:

$f (x) = \{\begin{matrix} \sqrt{- x}, i f x < 0 \\ 2, i f x = 0 \\ \sqrt{x}, i f x > 0 \end{matrix} a n d i t h a s i t' s b r e a k p o i n t a t x = 0 .$

Step 2. Finding the limits at the break point.

$A t x = 0, L H L = \lim_{x 鈫� 0^{-}} f (x) = \lim_{x 鈫� 0^{-}} \sqrt{- x} = \sqrt{- 0} = 0 . R H L = \lim_{x 鈫� 0^{+}} f (x) = \lim_{x 鈫� 0^{+}} \sqrt{x} = \sqrt{0} = 0 . f (0) = 2 . N o w, L H L = R H L 鈮� f (0) .$

Step 3. Finding the type of discontinuity.

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

Now we know LHL = RHL, none is equal to f(0) and both exist then this type of discontinuity is point discontinuity.

And also, $L H L 鈮� f (0) a n d a l s o R H L 鈮� f (0)$ So, it is neither left continuous nor right continuous.

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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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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,ifx<02,ifx=0x,ifx>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

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.