epsilon delta limit proof

prometheus54

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Mar 7, 2011
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Hi i need to prove the following. Use the ? - ? definition of the limit of a function to prove the lim(x->3) ((x^3)-4)/((x^2)-1)= 23/8.

So i started like this.
Let ? > 0. So there exists a ? > 0 such that for all x with 0< | x - 3 | < ? then |f(x) - (23/8) | < ?. So | f(x) - (23/8) | < ? iff |((x^3-4)/(x^2-1)) - (23/8) | < ?
So from there i kept factoring and eventually got | ( 8x^3- 23x^2 - 9)/ (8(x^2 - 1)) | < ?.....and now i am stuck, i do not know how to simplify anymore....any help would be appreciated. Thanks
 
Note that if delta < 1 then 2<x<4 so x^2-1 < 15. So your |f(x)/g(x)| <= 1/120|f(x)|

Then divide x-3 into your numerator and do the same thing for the other factor.
 
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