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Another delta, epsilon problem

  • #1
264
0

Homework Statement



For:

[tex] \lim_{x \to 1^{-}} \frac {1}{1-x^{2}} = \infty [/tex]

Find [tex]\; \delta > 0 \;[/tex] such that whenever:

[tex] 1-\delta<x<1 \;\;[/tex] then [tex] \; \frac {1}{1-x^{2}} > 100 [/tex]

Homework Equations



[tex] |x-a| < \delta [/tex]

[tex] |f(x)-L| < \epsilon [/tex]

The Attempt at a Solution



So as it is set right now, this:

[tex] \frac {1}{1-x^{2}} > 100 [/tex]

when solved for x will give me the upper limit but I am looking for the lower limit so I solve for x starting with this:

[tex] \frac {1}{1-x^{2}} < -100 [/tex]

[tex] 1-x^{2} > -10^{-2} [/tex]

[tex] -x^{2} > -10^{-2}-1 [/tex]

[tex] x^{2} > 10^{-2}+1 [/tex]

[tex] x > \sqrt{10^{-2}+1} [/tex]

and then since:

[tex] 1-\delta<x<1 = -\delta<x-1<0 [/tex]

then I will subtract 1 from both sides of the inequality to get:

[tex] x-1 > \sqrt{10^{-2}+1}-1 [/tex]

which works out to be:

[tex] x-1 > .0049 \; or \; \approx 5*10^{-3} [/tex]

which 5*10^-3 is the correct answer but I am not sure that I did the work correctly. They got an exact answer of:

[tex] \delta = 1 - \sqrt{.99} [/tex]

which is actually about .00504, so not my answer exactly

Someone help with where I went wrong?

I say that 5*10^-3 is correct because the answer in the back reads like so:

[tex] \delta = 1 - \sqrt{.99} \; or \; \approx 5*10^{-3} [/tex]

thanks!
 
Last edited:

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,538
1,149

Homework Statement



For:

[tex] \lim_{x \to 1^{-}} \frac {1}{1-x^{2}} = \infty [/tex]

Find [tex]\; \delta > 0 \;[/tex] such that whenever:

[tex] 1-\delta<x<1 \;\;[/tex] then [tex] \; \frac {1}{1-x^{2}} > 100 [/tex]

Homework Equations



[tex] |x-a| < \delta [/tex]

[tex] |f(x)-L| < \epsilon [/tex]

The Attempt at a Solution



So as it is set right now, this:

[tex] \frac {1}{1-x^{2}} > 100 [/tex]

when solved for x will give me the upper limit but I am looking for the lower limit so I solve for x starting with this:

[tex] \frac {1}{1-x^{2}} < -100 [/tex]
You don't want to do that because...
[tex] 1-x^{2} > -10^{-2} [/tex]

[tex] -x^{2} > -10^{-2}-1 [/tex]

[tex] x^{2} > 10^{-2}+1 [/tex]

[tex] x > \sqrt{10^{-2}+1} [/tex]
You have x>1 but you're trying to find the lower limit of x as you approach from the left.
 
  • #3
264
0
So are you saying I should have solved it as:

[tex] \frac {1}{1-x^{2}} > 100 [/tex]

instead of swapping it around? I thought that my way was correct because you end with the sign as something less than x and thus that would be the lower limit as it is less than x, is that incorrect logic?
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,538
1,149
Right, you shouldn't have swapped. You ended up solving the problem for x approaching 1 from the right.
 

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