# prove 0 to the infinite power is not indeterminate

#### 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.

#### HallsofIvy

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

#### pnfuller

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

#### Deveno

MHF Hall of Honor
here is a hint:

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

#### Plato

MHF Helper
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?

#### pnfuller

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

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

#### pnfuller

what do you mean?
here is a hint:

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

#### Plato

MHF Helper
what do you mean?
If $$\displaystyle g(x)\to\infty$$ then $$\displaystyle (0.5)^{g(x)}\to 0$$.

#### 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.
If $$\displaystyle g(x)\to\infty$$ then $$\displaystyle (0.5)^{g(x)}\to 0$$.
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 $$\displaystyle \lim _{x \to a} f(x) = L$$