Find the largest Delta

dangbeau · February 12th, 2013, 04:17 PM

How to solve this problem? can anyone help me
I'll be very appreciated if it's explained step by step

The graph of the following function is shown in the figure. Find the largest ? such that if 0 < |x � 2| < ? then |f(x) � 1| < 0.01.
f(x) =1/( x ? 1)

http://i.imgur.com/ZLvTR5h.png

Crimson Sunbird · February 13th, 2013, 04:41 AM

$\left|\frac{1}{x-1}-1\right|<\frac{1}{100}$

$\Leftrightarrow$ $-\frac{1}{100} < \frac{1}{x-1}-1 < \frac{1}{100}$

$\Leftrightarrow$ $\frac{99}{100} < \frac{1}{x-1} < \frac{101}{100}$

$\Leftrightarrow$ $\frac{100}{101} < x-1 < \frac{100}{99}$

$\Leftrightarrow$ $-\frac{1}{101} < x-2 < \frac{1}{99}$

Hence $\delta=\min\left\{\frac{1}{101}, \frac{1}{99}\right\}=\frac{1}{101}$ .

February 12th, 2013, 04:17 PM	#1
dangbeau Newbie Joined: Feb 2013 Posts: 1 Thanks: 0	Find the largest Delta How to solve this problem? can anyone help me I'll be very appreciated if it's explained step by step The graph of the following function is shown in the figure. Find the largest ? such that if 0 < \|x � 2\| < ? then \|f(x) � 1\| < 0.01. f(x) =1/( x ? 1) http://i.imgur.com/ZLvTR5h.png
$dangbeau is offline$

February 13th, 2013, 04:41 AM	#2
Crimson Sunbird Member Joined: Jan 2013 Posts: 93 Thanks: 0	Re: Find the largest delta $\left\|\frac{1}{x-1}-1\right\|<\frac{1}{100}$ $\Leftrightarrow$ $-\frac{1}{100} < \frac{1}{x-1}-1 < \frac{1}{100}$ $\Leftrightarrow$ $\frac{99}{100} < \frac{1}{x-1} < \frac{101}{100}$ $\Leftrightarrow$ $\frac{100}{101} < x-1 < \frac{100}{99}$ $\Leftrightarrow$ $-\frac{1}{101} < x-2 < \frac{1}{99}$ Hence $\delta=\min\left\{\frac{1}{101}, \frac{1}{99}\right\}=\frac{1}{101}$ .
$Crimson Sunbird is offline$

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