Limit of cos(x)^(1/x) - Solution Ideas

[a4swe]

Homework Statement

Find the limit:

\stackrel{lim}{x\rightarrow0} cos(x)^{1/x}

The Attempt at a Solution

As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?

 

[gb7nash]

Homework Statement

Find the limit:

\stackrel{lim}{x\rightarrow0} cos(x)^{1/x}

The Attempt at a Solution

As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?

What does the numerator tend to? What does the denomiator tend to?

Also, pay attention to the sign on the denominator. What can happen?

 

[C.Cochran]

Homework Statement

Find the limit:

\stackrel{lim}{x\rightarrow0} cos(x)^{1/x}

The Attempt at a Solution

As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?

Your intuition seems to be right, why is it wrong?
If its a stupid question i apologize, I'm working on learning Calculus.

 

[Office_Shredder]

Is it supposed to be
\lim_{x\to 0} \cos(x)^{1/x}
or
\lim_{x\to 0} \frac{\cos(x)}{x}