<s>[INST] 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:
$$
\lim_{n\to\infty}\frac{1}{e^n}e^{n+b+c/n}=
\lim_{n\to\infty}e^{n+b-n+c/n}=e^b
$$
 [/INST] I would say that $$\log_e\left(\frac{(\frac{1}{n}+1)^{bn+c+n^2}}{e^n}\right) $$  $$ = (bn+c+n^2)\log_e\left(\frac{1}{n}+1\right) - n$$  $$= (bn+c+n^2)\left(\frac{1}{n}-\frac{1}{2\,{n}^{2}}+\frac{1}{3\,{n}^{3}}-\frac{1}{4\,{n}^{4}}+\cdots\right) - n$$ $$= b - \frac{b}{2n} + o(1/n) +\frac{c}{n} - o(1/n) +n -\frac{1}{2} + \frac{1}{3n} -o(1/n) -n$$ $$= b  -\frac{1}{2} +\frac{-3b+6c+2}{6n} + o(1/n)$$
making the limit of the logarithm $b-\frac{1}{2}$ as $n\to \infty$ and so the original limit $\displaystyle e^{b-\frac{1}{2} }$.
</s>