Suppose I have $e^{(\frac{1}{n}b + o(\frac{1}{n}))}$ then $\lim_{n \rightarrow \infty} = e^0 = 1$ so $$e^{(\frac{1}{n}b + o(\frac{1}{n}))} = o(1) +1$$
But if I take the Taylor expansion of $e^{(\frac{1}{n}b + o(\frac{1}{n}))}$ I get that $$e^x = 1+\frac{1}{n}b + o(\frac{1}{n}) $$
Are these both correct(due to the misuse of notation one does not negate the other) ? In the cases when one can use the Taylor expansion does it always give better information?