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Chapter 1: Q. 72 (page 108)

Prove each of the limit statements in Exercises67鈥�72. You
will have to bound $未$ .
$\lim_{x 鈫� 3} \frac{18}{x^{2}} = 2$

Short Answer

Expert verified

The given limit $\lim_{x 鈫� 3} \frac{18}{x^{2}} = 2$ is proved.

Step by step solution

01

Step 1. Given Equation

$\lim_{x 鈫� 3} \frac{18}{x^{2}} = 2$

02

Step 2. Proving the limit

Given $蔚 > 0$ , choose $未 = m i n (1, \frac{蔚}{3})$ , Then if, $0 < | x - 3 | < 未$ we have

Or

role="math" localid="1652122774333" $蔚 > 0 a n d 未 > 0$

such that $| \frac{18 - 2 x^{2}}{x^{2}} | 鈮� 鈭�$ if $| x - 3 | 鈮� 未$

role="math" localid="1652123155385" $= | \frac{18 - 2 x^{2}}{x^{2}} | 鈮� 鈭� = \frac{2}{x^{2}} | (9 - x^{2}) | 鈮� 鈭� = \frac{2}{x^{2}} | (3^{2} - x^{2}) | 鈮� 鈭� = \frac{2}{x^{2}} | ((3 - x) (3 + x)) | 鈮� 鈭�$

when x=3

role="math" localid="1652123462932" $= \frac{2 | 3 + 3 |}{3^{2}} | (x - 3) | 鈮� 鈭� = \frac{2 | 6 |}{9} | x - 3 | 鈮� 鈭� = \frac{12}{9} | x - 3 | 鈮� 鈭� = \frac{9}{3} | x - 3 | 鈮� 鈭� = | x - 3 | 鈮� \frac{3}{4} 鈭�$

taking $未 = \frac{3 鈭�}{4}$ so that $| \frac{18}{x^{2}} - 2 | 鈮� 鈭� i f | x - 3 | 鈮� 未$

Hence Proved.

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Most popular questions from this chapter

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x = c$ .

$f (x) = \{\begin{aligned} 2 鈭� x, 鈥呪赌呪赌呪赌� if x rational \\ x^{2}, 鈥呪赌呪赌呪赌� if x irrational, \end{aligned} c = 1$

Write delta-epsilon proofs for each of the limit statements in Exercises $47 鈥� 60 .$

$\lim_{x 鈫� 2^{+}} 3 \sqrt{2 x - 4} = 0$

For each function f graphed in Exercises 23鈥�26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

Calculate each of the limits:

$\lim_{h 鈫� 0} \frac{{(2 + h)}^{2} - 2^{2}}{h}$ .

Write each of the inequalities in interval notation:

$|(3 x + 1) - 2| < 0.1$

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