Question 45
Let $f:\mathbb R\to\mathbb R$ be $n$ times differentiable, where $n\geq2$, and suppose that
$$
\lim_{x\to+\infty}f(x)=\ell\in\mathbb R,\qquad
\lim_{x\to+\infty}f^{(n)}(x)=0.
$$
Prove that
$$
\lim_{x\to+\infty}f^{(k)}(x)=0\qquad(k=1,2,\ldots,n-1),
$$
where $f^{(k)}$ denotes the $k$th derivative of $f$.
$$
\lim_{x\to+\infty}f(x)=\ell\in\mathbb R,\qquad
\lim_{x\to+\infty}f^{(n)}(x)=0.
$$
Prove that
$$
\lim_{x\to+\infty}f^{(k)}(x)=0\qquad(k=1,2,\ldots,n-1),
$$
where $f^{(k)}$ denotes the $k$th derivative of $f$.