In the limit $lim_(x->1^-)1/(1-x^2)=oo$ , how do you find $delta>0$ such that whenever $1-delta<x<1$ , $1/(1-x^2)>100$ ?

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Answer 1 · 2018-06-10T11:04:34

See below

It must be $1/(1-x^2)>100$

from here $1>100-100x^2$ then transposing terms we have

$100x^2>99$ then $x^2>99/10$ or equivalent $x>+-sqrt(99/100)$ .

we choose positive root because $delta>0$

$x>sqrt99/10$

If we take $delta=sqrt99/10$ we are sure that:

first.- $delta <1$
second.- if $x in (1-delta,1)$ then $1/(1-x^2)>100$

In the limit lim_(x->1^-)1/(1-x^2)=oolim_(x->1^-)1/(1-x^2)=oo, how do you find delta>0delta>0 such that whenever 1-delta<x<11-delta<x<1, 1/(1-x^2)>1001/(1-x^2)>100?