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