Question

Find the largest δ such that if 0 < |x – 1| < δ then |f(x) – 1| < 0.1?

Answer 1

\[f(x) = 2- \frac{ 1 }{ x }\]

Answer 2

what have you worked out so far?

Answer 3

@amistre64
I did f(x)=2-1/x
absolute value of 2-1/x < 0.1

0< absolute value of x-1< delta

1- 1/x < 0.1

Here's where I got stuck. Di I do it wrong?

Answer 4

we have to solve these two inequalities:
\[\Large \left| {1 - \frac{1}{x}} \right| < \frac{1}{{10}}\]

Answer 5

your starting fine, the idea is to convert the range into the domain, which is just algebra worked over ...

\[|f(x)-1|<.1\]
\[-.1<f(x)-1<.1\]
\[.9<f(x)<1.1\]
\[.9<2 - \frac 1x<1.1\]
\[.9-2<- \frac 1x<1.1-2\]
\[2-.9> \frac 1x>2-1.1\]
\[\frac1{2-.9}< x<\frac1{2-1.1}\]
\[\frac1{2-.9}-1< x-1<\frac1{2-1.1}-1\]

Answer 6

we have 2 deltas to try .... test them out to see what fits.

Answer 7

the smaller value is the safest bet to me ...

Answer 8

I've gotten -0.09 <x-1<0.11

Answer 9

-1/11 < x < 1/9

Answer 10

o sorry

Answer 11

\[2-\frac1{1-\frac1{11}}=1-.1\]
\[2-\frac1{1+\frac1{9}}=1+.1\]

Answer 12

but since 1/9 is greater than 1/11

we cant use x-1/9 and still be in our range of .9 to 1.1

Answer 13

\[2-\frac1{1-\frac1{9}}=1-.125\]

we are out of the bounds of our limitation

Answer 14

That'd give us 0.875 or 7/8

Answer 15

correct, which is smaller than .9

we need to remain within (.9, 1.1)

.875 < (.9, 1.1)

Answer 16

right

Answer 17

if we use the delta = 1/11 then the function fits in both cases. so its the largest value we can use.

Answer 18

which gives us 0.0909....

Answer 19

yes if they want a decimal representation of it

Answer 20

I checked it in my assignment online for that question, and 1/11 is the correct answer.

Answer 21

They take integers, fractions or decimals

Answer 22

well of course it is .. :)

Answer 23

@amistre64 thank you for the walk through. I just got stuck whiles working it out :)

Answer 24

youre welcome, ita just an exercise in algebra, converting range to the domain and picking the smaller result

Answer 25

u sir just got yourself a medal ;)

Answer 26

good luck :)

Answer 27

thank u