Prove Limits Using Epsilon-delta Definition...5

[nycmathguy]

Calculus
Section 2.4

I don't need to know this stuff to succeed in Calculus. However, I would like to have your math work reply file away as future study notes and/or reference notes. How about working out 24?

 

[MathLover1]

24.

Proof:
Let ϵ> 0 be given, now we need to choose a δ > 0 such that for
any x, that satisfies |x − a| < δ, the condition |f(x) − c| < ϵ is satisfied.

Well this one is an easy one to prove because f(x) = c for any x.
So |f(x) − c| = |c − c| = 0.

So there is no game here at all, because for any given ϵ> 0, it doesn’t matter what δ I choose, for any x, in any (a−δ, a+δ) interval corresponding |f(x) − c| = |c − c| = 0 < ϵ.

So in this case

.

 

[nycmathguy]

24. View attachment 3018

Proof:
Let ϵ> 0 be given, now we need to choose a δ > 0 such that for
any x, that satisfies |x − a| < δ, the condition |f(x) − c| < ϵ is satisfied.

Well this one is an easy one to prove because f(x) = c for any x.
So |f(x) − c| = |c − c| = 0.

So there is no game here at all, because for any given ϵ> 0, it doesn’t matter what δ I choose, for any x, in any (a−δ, a+δ) interval corresponding |f(x) − c| = |c − c| = 0 < ϵ.

So in this case View attachment 3017.

I yearn for the day this makes sense to me. Your reply is notgoing to waste. The goal is to study your reasoning to prove the limit. This I will do all in good time.