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

For part a, can we employ binomial theorem, take out first two therms and group everything else into small order notation?

For part a, could you do a first degree taylor series on (1+x)^c (only taking the first derivative and grouping everything else into o(x), or would that violate some theorem/assumption since we’re it to a polynomial instead of a non-polynomial function?