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,