Let

From the previous two exercises here and here (Section 10.4, Exercises #30 and #31) we know that

and that

Use these results to prove that

Then use the identity

to prove that

*Proof.* First,

*Proof.* Letting

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.