I'm trying to write a delta-epsilon proof that:

I know that exists if such that:

And exists if such that:

I don't understand how I can write a proof using an arbitrary function and limit. So far I have only been proving specific limits with the epsilon delta method, not writing any algebraic proofs like this. Can someone offer some help? Thanks

Edit: I had a thought, but I'm not sure I wrote that second one out right, cause wouldn't I simply have to prove: (which isn't true)?