Q:

Explain why $\lim ({a_n}^k)={(\lim(a_n))}^k$ cannot be used to find limit $(1+\frac1n)^n$

Explain why lim $({a_n}^k)=(\lim(a_n))^k$ cannot be used to find limit $(1+\frac1n)^n$
The condition for this result to hold is $(a_n)$ has to be a convergent sequence and $k \in \mathbb N$, which are both true in this case, because lim $(1+\frac1n) =1$. So why can't we use this?

A:

Because n is variable, while k is a constant in your property.
What means $(\lim_{n\to +\infty} a_n )^n$? $n$ is both inside the limit and outside, so what is the value of $n$ outside? So this doesn't make sense syntaxically

