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Q. 88

Expert-verified

Calculus

Found in: Page 122

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Write a delta–epsilon proof that proves that f is continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1.
$f (x) = x^{- 1}$

Ans: $f (x) = x^{- 1}$ is continuous on its domain (continuous for all role="math" localid="1648040276972" $x \in R - {0}$ )

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given, $f (x) = x^{- 1}$

Step 2. Find Domain.

$f (x) = x^{- 1} = \frac{1}{x}$

At $x = 0$ ,

$f (0) = \frac{1}{0} = \infty$

Hence, is not defined at localid="1648042677677" role="math" $x = 0$

Step 3. So, check for continuity at all points except 0.

Let c be any real number except $0 .$

assume that $c$ is less than or equal to $1$

$f$ is continuous at $x = c$

if, $lim_{x \to c} f (x) = f (c)$

localid="1648049491586" $L H S = lim_{x \to c} f (x) = lim_{x \to c} \frac{1}{x} P u t t i n g x = c = \frac{1}{c}$

localid="1648049934543" $R H S = f (c) = \frac{1}{c}$

Step 4. LHS=RHS

The function is continuous at $x = c$ (Except $0$ )

Thus, we can write that

$f i s c o n t i n u o u s f o r a l l x \in R - {0}$

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