I cannot solve this limit $$
\lim_{n\to\infty}\frac{(\frac{1}{n}+1)^{bn+c+n^2}}{e^n}=e^{b-\frac{1}{2}}
$$
I am doing it like this, and I cannot find the mistake:
$$
\lim_{n\to\infty}\frac{1}{e^n}e^{n+b+c/n}=
\lim_{n\to\infty}e^{n+b-n+c/n}=e^b
$$

hint: $a_n=\left(\dfrac{\left(1+\frac{1}{n}\right)^n}{e}\right)^n \Rightarrow \ln (a_n) = \dfrac{\ln(1+\frac{1}{n})-\frac{1}{n}}{\frac{1}{n^2}}\to 0$ by L'hospitale rule with $x = \frac{1}{n} \to 0 \Rightarrow a_n \to 1$. Given that the answer is $e^{b-1/2}$, can you figure the other part ?
