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. 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.