# Thread: prove 0 to the infinite power is not indeterminate

1. ## prove 0 to the infinite power is not indeterminate

suppose f is a positive function. if limx->a f(x) = 0 and limx->a g(x) = infinity, show that

limx->a [f(x)]g(x) = 0

this shows that 0 to the infinite power is not an indeterminate form.

2. ## Re: prove 0 to the infinite power is not indeterminate

What does "$\displaystyle \lim_{x\to a} f(x)= 0$" mean? And what does "[tex]\lim_{x\to a} g(x)= \infty[/itex]" mean?

3. ## Re: prove 0 to the infinite power is not indeterminate

this is where the problem came from
Originally Posted by HallsofIvy
What does "$\displaystyle \lim_{x\to a} f(x)= 0$" mean? And what does "[tex]\lim_{x\to a} g(x)= \infty[/itex]" mean?

4. ## Re: prove 0 to the infinite power is not indeterminate

here is a hint:

show log(f(x))*g(x) is very large negative, for x sufficiently close to a.

5. ## Re: prove 0 to the infinite power is not indeterminate

Originally Posted by pnfuller
suppose f is a positive function. if limx->a f(x) = 0 and limx->a g(x) = infinity, show that
limx->a [f(x)]g(x) = 0
this shows that 0 to the infinite power is not an indeterminate form.
There is an open interval $\displaystyle (a-\delta,a+\delta)$ on which $\displaystyle |f(x)|<0.5~\&~g(x)>1$.

So there $\displaystyle |f(x)|^{g(x)}<(0.5)^{g(x)}$.

Now as $\displaystyle \delta\to 0^+$, what does that tell you?

6. ## Re: prove 0 to the infinite power is not indeterminate

but how can you do that if the log(f(x)) is undefined at 0?
Originally Posted by Deveno
here is a hint:

show log(f(x))*g(x) is very large negative, for x sufficiently close to a.

7. ## Re: prove 0 to the infinite power is not indeterminate

what do you mean?
Originally Posted by Deveno
here is a hint:

show log(f(x))*g(x) is very large negative, for x sufficiently close to a.

8. ## Re: prove 0 to the infinite power is not indeterminate

Originally Posted by pnfuller
what do you mean?
Look at reply #5.
If $\displaystyle g(x)\to\infty$ then $\displaystyle (0.5)^{g(x)}\to 0$.

9. ## Re: prove 0 to the infinite power is not indeterminate

but i have to show that using a definition or something, my teacher says using numbers as examples to prove something really isnt a proof.
Originally Posted by Plato
Look at reply #5.
If $\displaystyle g(x)\to\infty$ then $\displaystyle (0.5)^{g(x)}\to 0$.

10. ## Re: prove 0 to the infinite power is not indeterminate

Originally Posted by pnfuller
but i have to show that using a definition or something, my teacher says using numbers as examples to prove something really isnt a proof.
Tell your teacher that this is a well known principle.
If $\displaystyle h(x)\le f(x)\le g(x)$ and $\displaystyle \lim _{x \to a} h(x) = \lim _{x \to a} g(x)=L$
then $\displaystyle \lim _{x \to a} f(x) = L$

11. ## Re: prove 0 to the infinite power is not indeterminate

i can't read post 5 and the weird little symbols it has

12. ## Re: prove 0 to the infinite power is not indeterminate

Originally Posted by pnfuller
i can't read post 5 and the weird little symbols it has
If that is true, I would say that the original question is over you head.
It is a higher level concept question.
To answer it properly one needs higher level understanding of limits than you seen to have.

13. ## Re: prove 0 to the infinite power is not indeterminate

its an extra credit problem and i need the extra credit so i was really trying to figure it out!
Originally Posted by Plato
If that is true, I would say that the original question is over you head.
It is a higher level concept question.
To answer it properly one needs higher level understanding of limits than you seen to have.

14. ## Re: prove 0 to the infinite power is not indeterminate

Originally Posted by pnfuller
its an extra credit problem and i need the extra credit so i was really trying to figure it out!
If I were you, I would say it is a known fact that:
if $\displaystyle |r|<1$ then $\displaystyle \lim _{x \to \infty } r^x = 0$.

Because $\displaystyle \lim _{x \to a } f(x) = 0$ then $\displaystyle 0\le |f(x)|<0.5$ for all $\displaystyle x$ near $\displaystyle a$.

But we are given $\displaystyle \lim _{x \to a } g(x) = \infty$ so $\displaystyle (0.5)^{g(x)}\to 0$.

So $\displaystyle \lim _{x \to a} f(x)^{g(x)} = 0$ because $\displaystyle 0\le|f(x)|^{g(x)}\le (0.5)^{g(x)}\to 0$

Note that $\displaystyle |f(x)|^{g(x)}=|f(x)^{g(x)}|$

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