prove 0 to the infinite power is not indeterminate

Aug 2012
107
0
north carolina
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
Apr 2005
20,249
7,909
What does "\(\displaystyle \lim_{x\to a} f(x)= 0\)" mean? And what does "\(\displaystyle \lim_{x\to a} g(x)= \infty[/itex]" mean?\)
 
Aug 2012
107
0
north carolina
this is where the problem came from IMG_4226.jpg
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
Mar 2011
3,546
1,566
Tejas
here is a hint:

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

Plato

MHF Helper
Aug 2006
22,507
8,664
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?
 
Aug 2012
107
0
north carolina
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.
 
Aug 2012
107
0
north carolina
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.
Look at reply #5.
If \(\displaystyle g(x)\to\infty\) then \(\displaystyle (0.5)^{g(x)}\to 0\).
 

Plato

MHF Helper
Aug 2006
22,507
8,664
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 \(\displaystyle \lim _{x \to a} f(x) = L\)