Consider the calculation of the following limit: $$\lim_{n\rightarrow\infty}\frac{1}{e^{n^2x}}$$
I did these passages... Could you tell me if the whole resolution is formally right?
$$\lim_{n\rightarrow\infty}\frac{1}{e^{n^2x}} = \frac{\lim_{n\rightarrow\infty} 1}{\lim_{n\rightarrow\infty} e^{n^2x}}$$
(This is the passage I'm most uncertain of...)
I try to apply the famous limit: $\lim_{n\rightarrow\infty} 1^n = 1$, so I substitute 1 with $1^n$
$$= \frac{\lim_{n\rightarrow\infty} 1^n}{\lim_{n\rightarrow\infty} e^{n^2x}}$$
I do substitute 1 with $1^n$ again
$$\frac{\lim_{n\rightarrow\infty} (1^n)^n}{\lim_{n\rightarrow\infty} e^{n^2x}} = \lim_{n\rightarrow\infty} \frac{1^{n^2}}{e^{n^2 x}} = \lim_{n\rightarrow\infty} (\frac{1}{e^x})^{n^2}$$
At this point I substitute $n^2 = n'$ and study the base of the power ($\frac{1}{e^x}$) at the varying of x, calculating the easy limit $\lim_{n'\rightarrow\infty} (\frac{1}{e^x})^{n'}$for $x<0, x=0$ and $x>0$
It's all correct or there are some errors/imprecisions in the calculation? I'm very uncertain on the fact that I can, formally, substitute $1^n$ to $1$ only because of $\lim_{n\rightarrow\infty} 1^n = 1$... Is it possible to substitute one expression with another only because they have the same limit?...