Delta epsilon

how would i prove that lim of x^4 as x->p is p^4? x^4-p^4 = (x-p)(x+p)(x^2+p^2). I'm having trouble controlling the (x+p)(x^2+p^2) term without having to resort to proving for p > 0 and p < 0 seperately.

shmoe
Homework Helper
A standard idea is to place an upper bound on your delta to restrict x, then use this to bound the (x+p)(x^2+p^2) part. If you knew delta<=|p|, can you find an upper bound for |(x+p)(x^2+p^2)|?

This is assuming p is not 0. You could modify the delta above to something that would work in this case, or just deal with it seperately.

I get |x+p| < |3p| and |x^2+p^2| < 5p^2 if delta < |p|. So then 15p^3 is an upper bound. Is this correct? And as far as dealing with p = 0, can I just add one to the denominator? How did you come up with using |p| for delta?

shmoe