# Math Help - Deriving the equation

1. ## Deriving the equation

Hi all!

I have the following general equation which is: $s(s+t)=((1-t)s(x)^\alpha + ts(x+1)^\alpha)^\frac{1}{\alpha}$. where s is survival function

It is told to me that when I substitute $\alpha =0$, I will be able to obtain

$log(s(x+t))=(1-t)log (s(x))+(t) log (s(x+1))$.

But how do I do that since when $\alpha=0$, $1/\alpha$ goes to infinity??

2. Originally Posted by noob mathematician
Hi all!

I have the following general equation which is: $s(s+t)=((1-t)s(x)^\alpha + ts(x+1)^\alpha)^\frac{1}{\alpha}$. where s is survival function

It is told to me that when I substitute $\alpha =0$, I will be able to obtain

$log(s(x+t))=(1-t)log (s(x))+(t) log (s(x+1))$.

But how do I do that since when $\alpha=0$, $1/\alpha$ goes to infinity??
Since direct substitution leads to the indeterminant form $1^{\infty}$ you would be expected to take the limit $\alpha \to 0$.

3. Originally Posted by mr fantastic
Since direct substitution leads to the indeterminant form $1^{\infty}$ you would be expected to take the limit $\alpha \to 0$.
Thanks for the hint.
But I still finding it hard to take the limit to 0. How do I go about doing it? Substitute $\alpha$ with a very small number? 0.000001 etc
Or do I differentiate the equation?

4. Originally Posted by noob mathematician
Thanks for the hint.
But I still finding it hard to take the limit to 0. How do I go about doing it? Substitute $\alpha$ with a very small number? 0.000001 etc Mr F says: NO. That's not how you take a limit.
Or do I differentiate the equation?
The technique can be found here: Indeterminate form - Wikipedia, the free encyclopedia

5. Oh great! L'Hôpital's rule.. Thanks so much

6. take the logarithm first