What is $\lim_{x \to \infty} 0^{\frac{1}{x}}$? Is it 0 or 1 or is it an invalid limit? I’m unsure and lack the tools to find out if $\lim_{x \to \infty} 0^{\frac{1}{x}}$ is 0 or 1 or is it an invalid limit. One way $0^0$ feels like it should give 1 but then again, the function is 0 for all $x$. So, how do we talk about such a limit? Also, does the limit remain the same when we replace $x$ by the natural number $n$. Please provide proper reasoning in your answers.
Note: Wolfram Alpha says that it is zero.

This function is identically zero in $(0,\infty)$. For any positive $x$ we have $0^{\frac{1}{x}}=0$. So of course the limit in that case is zero, you can even easily prove this by the definition of the limit. 
