# One raised to the infinity power help please

1. Sep 26, 2008

### arpitm08

one raised to the infinity power help please!!!

Let a and b be positive real numbers. For real number p define, f(p) = ((a^p + b^p)/2)^(1/p). Evaluate the limit of f(p) as p approaches 0.

By directly plugging in zero, you would get (1)^inf. Wouldn't that equal 1 or would it be something else? When I put it into my 89 i got 1 as my answer, but for some reason I don't think that it is right. Please help!

2. Sep 26, 2008

### 00PS

Re: one raised to the infinity power help please!!!

why wouldn't it? 1x1x1x1... is still 1 isn't it?

you can also check the graph at 0 to see if it converges to 1 when you plug in positive real numbers for a and b. don't the 89s take limits?

3. Sep 26, 2008

### sutupidmath

Re: one raised to the infinity power help please!!!

$$1^{\infty}$$ is undefined!

4. Sep 26, 2008

### arildno

Re: one raised to the infinity power help please!!!

If a=b, note that the limit is "a".

Let then "a" be greater than "b", and rewrite:
$$(\frac{(a^{p}+b^{p}}{2})^{\frac{1}{p}}=\frac{a}{2^{\frac{1}{p}}}({1+\frac{1}{N})^{\frac{1}{p}}, \frac{1}{N}=(\frac{b}{a})^{p}$$
Then,
$$\frac{1}{p}=N\frac{\ln(\frac{a}{b})}{N\ln(N)}$$

Therefore, you may rewrite this as:
$$((1+\frac{1}{N})^{N})^{\frac{\ln(\frac{a}{b})}{N\ln(N}}}$$
which remains nasty..

Last edited: Sep 26, 2008
5. Sep 26, 2008

### lurflurf

Re: one raised to the infinity power help please!!!

"If a=b, note that the limit is "a"."
Indeed.
"Wouldn't that equal 1"
No.
That is what is called an indeterminite form.
further analysis is needed.
f(p)=((a^p + b^p)/2)^(1/p)
Let us consider an approximation that is exact in the limit.
a^p~1+p log(a)
b^p~1+p log(b)
if p~0
thus
(a^p + b^p)/2~1+p log sqrt(ab)
if p~0
(1+x)^(1/P)~exp(x/p)
if p~0 and x~0

by combining these the answer should be clear.

6. Sep 26, 2008

### Anthony

Re: one raised to the infinity power help please!!!

Hint: if your sequence is $$f(p)$$, consider the limit of the related sequence $$g(p)$$ where $$g=\log f$$. Then use some standard properties of continuity.