Precise Def of Limit

[ku1005]

Hey guys, im new to this topic and just wodering how my lecturer got the folowing, as he said it was easy to do ourselves??

(e=epsilon)


"show that the limit x-> (infinite) of (x^2-1)/(x^2+1)=1"

specifically, he just comes right out and says let K = sqrt(2/e) and let x>K

impling 2/x^2 < e etc etc

I understand that letting K = above, means the equation ends up nicely becoming less then epsilon, however, what is the systematic approach to solving this value????...ie so i can use it for other equations which it is not as obvious?...if a process exists that is...

i tried this

-e < ( x^2-1)/(x^2-1) - 1 < e

-e < -2/(x^2+1) < e

-e < 2/(x^2+1) < e

but then how can you say that 2/(x^2+1)<=2/x^2<e ?????this the part I dont get!!!


so any help would be gretaly appreciated!!!

cheers

rhys

 

[Unco]

x^2 + 1 >= x^2 so of course 1/(x^2 + 1) =< 1/(x^2).

Consider the stage where you have |2/(x^2 + 1)| < e. Then 2/(x^2) < e, or x^2 > 2/e so x < -sqrt(2/e) or x > sqrt(2/e). Thus an appropriate K>0 would be sqrt(2/e).

This is the systematic approach, if you like.