Exponent can't be pulled from limit

Question

Using the limit property $$\lim_{x \to a}[f(x)^{n}] = \left(\lim_{x \to a}[f(x)]\right)^{n},$$ it seems to me that $$\lim_{x \to \infty}\left[\frac{1}{x^{-1}}\right] = \left(\lim_{x \to \infty}\left[\frac{1}{x}\right]\right)^{-1}$$ but this would equal $0^{-1}$ which is undefined. I'm wondering why the exponent limit property doesn't seem to hold in this case. Thanks.

Answer 1

The limit property says: if $\lim\limits_{x\to a}f(x)=L$, then $\lim\limits_{x\to a} \left[f(x)\right]^n=\left[\lim\limits_{x\to a}f(x)\right]^n=L^n$.

Since your limit does not exist, you will not get anything meaningful by rewriting the expression.

Answer 2

This requires that both of the limits actually exist. In this case however, $$\lim_{x \to \infty} \frac{1}{x^{-1}} = \lim_{x \to \infty} x$$ does not exist.