Limits misconception

Calculus Level 3

If $f:\mathbb{R}\to\mathbb{R}$ , is the following true or false? $\left(\forall m\in\mathbb{N}, \lim_{x\to\infty}\dfrac{d^mf(x)}{dx^m}=0\right)\iff\left(\exists a\in(-\infty,\infty):\lim_{x\to\infty}f(x)=a\right)$

$\mathbb{N}$ is the set of natural numbers .
$\mathbb{R}$ is the set of real numbers .

True False

1 solution

Piotr Idzik
Jan 9, 2021

Consider a smooth function $f \colon \R \to \R$ such that $f(x) = \sqrt{x}$ , for all $x > M$ , for some $M > 0$ . Then

$\lim_{x \to \infty}$ f(x) does not exist in $\R$ ,
for every $m \in \N$ , $\lim_{x \to \infty}\frac{\mathrm{d}^mf(x)}{\mathrm{d}x^m} = 0$ .

Limits misconception

1 solution

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