Limit problem involving x to the power of a function

[Wormaldson]

Homework Statement

Calculate the limit \lim_{x\rightarrow +\infty} x^{\frac{1}{x^2}}

2. The attempt at a solution

At first I was tempted to rewrite the function as a composition and go from there. I soon realized that this wouldn't work as the term being raised to the power of a function of x was a different function of x; to my knowledge, decomposing functions in this way will only work for functions like a^{f(x)}So no luck there.

The next thing I tried was manipulating the function algebraically; unfortunately I just ended up going in circles and wasn't able to get the function into a more useful form.

I came up with another method that seems superficially plausible, but I'm not sure if it's actually valid: that is, rewriting the problem as x^{\lim_{x\rightarrow +\infty} {\frac{1}{x^2}}}Intuitively, this seems sensible; the limit at infinity of the exponent is 0 and any x > 0 raised to the power of 0 is 1. The course book mentions that you can take a limit "inside" a continuous function (the exact wording). Obviously the function in question isn't continuous on it's entire domain, but it is continuous for all x > 0. Unfortunately, however, the course book doesn't go into any greater detail, and the textbook doesn't address limits of this type at all. If taking the limit "inside" the function in such a manner is a valid operation, then I guess my problem is solved. If not, then I could use a suggestion as to what to try next.

As always, any help is much appreciated.

 

[LCKurtz]

If $y = x^\frac 1 {x^2}$, try taking the limit of $\ln y$.

 

[ehild]

Using that x=e^ln(x), the function can be written in the form (e^{\ln(x)}) ^{\frac{1}{x^2}}=e^{\ln(x)\frac{1}{x^2}}

ehild