Examples of Finite and Infinite Values for 0^∞ Indeterminate Form?

[Mentallic]

Not so much a homework problem as a curiosity on my part. I chose to give a presentation recently on undefined numbers. With that, indeterminate's unsurprisingly found their way into my presentation.

After reading up on the list of indeterminate forms, I stumbled upon the form $$0^\infty$$ and for the life of my couldn't think of any examples in limits that have this form.

In my mind, I see such indeterminates as $$1^\infty$$ as trying to say
"multiplying 1 by itself repeatedly obviously still gives 1, but we're trying to do it so many times that it finally equals something other than 1".
Such an example would be e:

$$\lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}$$

Now a quick example of $$0^\infty$$ would be $$\lim_{x\rightarrow \infty}\left(\frac{1}{x}\right)^x$$

but in a way, I see this as "enforcing" the answer zero since this limit tends to zero much faster than $$\lim_{x\rightarrow \infty}\frac{1}{x}$$ does.

So can anyone give me an example of such an indeterminate form that equals a finite, and possibly even infinite value.

 

[Tedjn]

Where did you find the reference to $0^\infty$ as an indeterminate form? Off the top of my head, I can't think of a reason why this should be indeterminate.

 

[cronxeh]

Why is $1^\infty$ indeterminate?

Nevermind found it in the other thread.

x= 1^infinity is equivalent to ln(x)= infinity*ln(1)= infinity*0=0/1/infinity = 0/0

So is 0^infinity then equivalent to ln(x) = infinity*ln(0) = infinity*-infinity

 

[jack action]

According to http://en.wikipedia.org/wiki/Indeterminate_form" , it is not an indeterminate form.

The explanation being under the paragraph http://en.wikipedia.org/wiki/Indeterminate_form#Undefined_forms_that_are_not_indeterminate".