I've been reading through Spivak's calculus, and the problem is the answer key i have a hold of is for a different edition so it often doesn't answer the correct questions.(adsbygoogle = window.adsbygoogle || []).push({});

Anyways, here they are:

Chapter 5 problem 10

b. Prove that lim x-> 0 f(x) = lim x-> a f(x-a)

c. Prove that lim x-> 0 f(x) = lim x-> 0 f(x^3)

I'm mainly having a problem with making my solution rigorous.

For b what I have is on the left side we know there is a α_1 for every ε so

|x| < α_1 and |f(x)-m| < ε

We can choose the same epsilon for the right side:

|x-a| < α_2 and |f(x-a)-n| < ε

So the two x's are necessarily related so I replace it in the second equation with y and I also choose α=min(α_1,α_2).

So we have:

|x| < α and |f(x)-m| < ε

|y-a| < α and |f(y-a)-n| < ε

So now I want to say something like, let y-a=x so if ε<|m-n|/2 then it is impossible for both to be fulfilled unless m=n but I'm not sure how to say that rigorously.

I have a similar problem with the next problem.

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# Real analysis epsilon delta problem

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