Prove the formula for the limit:

Using,

* Proof. * We know from the previous exercise (Section 3.6, #17),

Then,

But we know from another earlier exercise (Section 3.6, #15) that

So, we have,

Hence, by Apostol’s Theorem 3.1 (ii), the limit of the difference is the difference of the limits:

use the identity (sin(5x)-sin(3x))=2cos((5x+3x)/2)sin((5x-3x)/2)=2cos(4x)sin(x)