Prove the formula for the limit:

Using,

(** Note: ** I think there is an error in this problem as stated in Apostol. The statement in the book is for the limit as . However, the limit as of this is not (since it is a quotient of continuous functions and for the denominator is nonzero at ). I’ve changed the statement of the question to be the limit as approaches . This makes the given formula correct, so is probably what was intended.)

* Proof. * From Theorem 2.3 (g) (Apostol, p. 96) we know

So, evaluating the limit we have,

But then,

where,

So,

Why are we allowed to change the variable of limits via substitution? I know it makes intuitive sense but given the rigorous treatment of everything else, this claim seems quite unsubstantiated

I have the same concern, and it has been raised before. See: https://math.stackexchange.com/questions/167926/formal-basis-for-variable-substitution-in-limits

Thanks, I thought I was going crazy 🤪. By the way, the limit when x->0 is sin(a)/a, no minus sign. Cheers.