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
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, 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?