Prove that

Using this fact compute

*Proof.* We use the definition and continuity of the exponential to evaluate the limit,

Then, we know (Examples on page 287 of Apostol, taking ) that

Therefore,

Finally we use the expansion, , and Theorem 7.8 (page 288 of Apostol on the algebra of the -symbols) to conclude,

Now, to use this to evaluate the requested limit we make the substitution . Then as and we have