I've been thinking about and trying to solve the following limit that I just feel lost by now. I always get an indeterminate form. I don't know what else to try. In the picture is just one way that I tried to do it, again resulting in an indeterminate:
Can you help?
$lim_{n\to \infty} \left[ \left(3+\dfrac{1}{n} \right)^{-3n} * 27^n\right]$
On
My suggestion was to notice that $$ \left(a+\frac1n\right)^n = \left(a+\frac{a}{an}\right)^n = a^n\left(1+\frac1{an}\right)^n \sim a^ne^{1/a}. $$
You can compute your limit as $$ \left(3+\frac1n\right)^{-3n}3^{3n} = 3^{-3n}\left(1+\frac1{3n}\right)^{-3n}3^{3n}=\left(1+\frac1{3n}\right)^{-3n} \to e^{-1}. $$
Let $x=\dfrac{t}{3}$ then your limit will be $$\lim_{t\to\infty}\left(1+\dfrac{1}{t}\right)^{-t}=\dfrac{1}{e}$$