## upper bounded taylor

$f(x)$ is two times diff. function on $(0, \infty)$ .
$\lim\limits_{x\rightarrow \infty} f(x) = 0$ satisfy.
$M=\sup\limits_{x>0}\vert f^{\prime \prime} (x) \vert$ satisfy
. for each integer $L$ ,
$g(L) = \sup\limits_{x\geq L} \vert f(x) \vert$, and $h(L) = \sup\limits_{x\geq L} \vert f^{\prime}(x) \vert$. for any $\delta > 0$, SHOW

$h(L) \leq \dfrac{2}{\delta} g(L) + \dfrac{\delta}{2}M$.