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Q. 43

Expert-verified

Calculus

Found in: Page 107

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.
$\lim_{x \to 1^{-}} \frac{1}{1 - x} = \infty$ , find $δ$ in terms of $M$ .

For $\lim_{x \to 1^{-}} \frac{1}{1 - x} = \infty$ , $δ = \frac{1}{M}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

$\lim_{x \to 1^{-}} \frac{1}{1 - x} = \infty$ .

Step 2. From the limit expression,

$f (x) = \frac{1}{1 - x} c = 1 L = \infty$

Now for $δ > 0$ .

$|x - 1| < δ |1 - x| < δ \frac{1}{|1 - x|} < δ |\frac{1}{1 - x}| < δ < M$

Hence, $δ = \frac{1}{M}$ .

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Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

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(e) If the graph of $f$ has a vertical asymptote at $x = 5$ , then $\lim_{x \to 5} f (x) = \infty$ .

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(g) If $\lim_{x \to 2} f (x) = \infty$ , then the graph of $f$ has a horizontal asymptote at $x = 2$ .

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