
Function
Hi everybody,
I must show that there exists a function f defined on $\displaystyle [1,+\infty[$ such as :
$\displaystyle \{{(\forall x \in [1,+\infty[) \sqrt{1+x}=1+\frac{1}{2}x\frac{1}{8}x^2+x^2\phi(x)\atop\lim_{x\to 0} \phi(x)=0}$
I don't know what to do, can you help me please.
And thank you anyway.

Hi
Let $\displaystyle \phi(x)=\frac{ \sqrt{1+x}\left(1+\frac{1}{2}x\frac{1}{8}x^2\right)}{x^2}$
$\displaystyle \phi$ is defined over $\displaystyle [1,0[ U ]0,+\infty[$
You just need to show that $\displaystyle \lim_{x\to 0} \phi(x)=0$