# Function

• Aug 17th 2009, 12:18 PM
lehder
Function
Hi everybody,

I must show that there exists a function f defined on $[-1,+\infty[$ such as :

$\{{(\forall x \in [-1,+\infty[) \sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^2+x^2\phi(x)\atop\lim_{x\to 0} \phi(x)=0}$

I don't know what to do, can you help me please.

And thank you anyway.
• Aug 17th 2009, 01:08 PM
running-gag
Hi

Let $\phi(x)=\frac{ \sqrt{1+x}-\left(1+\frac{1}{2}x-\frac{1}{8}x^2\right)}{x^2}$

$\phi$ is defined over $[-1,0[ U ]0,+\infty[$

You just need to show that $\lim_{x\to 0} \phi(x)=0$