Notation about which I don't fully understand What is the meaning behind writting $$\lim_{x\rightarrow a}e^{f(x)}=e^{\lim{x\rightarrow a}f(x)}$$ Has it something to do with continuity of the exponential? If so, if we had a limit like the following $$\lim_{x\rightarrow \infty}\Bigg[\Big(1+\frac{1}{x}\Big)^x\Bigg]^{f(x)}$$
does this property explain why we can evaluate a $1^{\infty}$-limit "by parts", by first seeing that $$\lim_{x\rightarrow \infty}\Big(1+\frac{1}{x}\Big)^x=e$$ and that $$\lim_{x\rightarrow \infty}f(x)=L$$ and so we can deduce that $$\lim_{x\rightarrow \infty}\Bigg[\Big(1+\frac{1}{x}\Big)^x\Bigg]^{f(x)}=e^L$$
 A: In this answer I try to address the question of the OP and the question of Dave.
In the OP's question it is written
$$
\lim_{x \rightarrow a} e^{f(x)} = e^{\lim_{x \rightarrow a} f(x)}
$$
and from this I infer that $\lim_{x \rightarrow a} f(x)$ is well defined as a real (maybe $\pm \infty$ if you wish) number. Then the fact that the above equality holds is exactly because the function $x \mapsto e^x$ is continuous. Indeed, because the exponential is continuous, whenever $f$ has a limit at $a$ by composition of limits the compound map $x \mapsto e^{f(x)}$ has a limit at $a$, and the above formula gives the expression of the limit.
Then the OP wishes to evaluate
$$
\lim_{x \rightarrow \infty} \left[ \left( 1 + \frac 1 x \right)^x \right] ^{f(x)},\quad L = \lim_{x \rightarrow \infty} f(x) \in \mathbb R
$$
that is not accurately called $1^\infty$. Here it is more like $e^L$, which is wrong in general. For instance try
$$
f(x) = 1/x,\quad f(x) = 1/x^2.
$$
My comment about the notation $e^x$ is motivated by the following observation: to define the exponentiation $x^y$ we use the formula
$$
x^y = \exp(y \ln(x)).
$$
So trying to represent $e^x$ by powers is (in my opinion) always misleading because
$$
\left( 1 + \frac 1 x \right)^x := \exp \left( x \ln \left( 1 + \frac 1 x \right) \right).
$$
Said differently, if one wishes to express $e^x$ by approximating the number $e$, say by a sequence $(e_n)$, and then computing $\lim_n e_n^x$ I would say this is nonsens because computing $e_n^x$ relies on the knowledge of the function $\exp$. Moreover a more sophisticated approximation
$$
\lim_{x \rightarrow \infty} \left[ \left( 1 + \frac 1 x \right)^x \right] ^{f(x)} = e^{\lim_{x \rightarrow \infty} f(x)}
$$
was proved to be wrong. Therefore I suggest that in this type of situation one may think twice before to write $e^x$. Nonetheless it is by definition that
$$
e^x = \exp(x)
$$
and with that I could not differ.
A: It is assumed that the limit of the function converges. Then you could treat it as any regular constant. Note that this works only when the limit converges.
