From the previous two exercises here and here (Section 10.4, Exercises #30 and #31) we know that
Use these results to prove that
Then use the identity
to prove that
we have from Exercise #31,
Then by Exercise #30,
Using the identity we then have
What if i wanted to proof both of those proofs with the definition of limit? Is there any way?
Yes, there is a proof in the continuous-function section. It can be modified to prove this one with the basic definition through special cases.