Chapter 1: Q. 55 (page 98)

For each limit in Exercises 55鈥�64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.
$\underset{x 鈫� 1^{+}}{l i m} \frac{1}{x^{2} - 1} = 鈭�, M = 1000, find largest 未 > 0$

Short Answer

Expert verified

The required value of $未鈮� 0.0005$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \frac{1}{x^{2} - 1}$

Step 2. Explanation

From the given function, we have, $c = 1, M = 1000$

The limit expression can be written as a formal statement as below,

For all M positive, there exists a delta positive such that if localid="1648046526661" $x 鈭� (1, 1 + 未)$

Then localid="1648047018883" $\frac{1}{x^{2} - 1} 鈭� (1000, 鈭�)$

Now the largest value of delta is given by,

localid="1648046734126" $\frac{1}{x^{2} - 1} = 1000 x^{2} - 1 = \frac{1}{1000} x^{2} = 0.001 + 1 x = \sqrt{1.001}$

Hence,

localid="1648033379330">未=1.001-1鈮�1.000499-1鈮�0.000499鈮�0.0005

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For each limit in Exercises 55鈥�64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.limx鈫�1+1x2-1=鈭�,M=1000,findlargest未>0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.