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What can you say about f(x)

Source: MTRP 2019 Class 11-Multiple Choice Question: Problem 1 :-

April 9, 2020
function

Problem Statement

Let f:(0,∞)→Rf : (0, \infty) \to \mathbb{R} is differentiable such that lim⁡x→∞f(x)=2019\lim \limits_{x \to \infty} f(x)=2019 Then which of the following is correct?
[*] lim⁡x→∞f′(x)\lim \limits_{x \to \infty} f'(x) always exists but not necessarily zero. [*] lim⁡x→∞f′(x)\lim \limits_{x \to \infty} f'(x) always exists and is equal to zero. [*] lim⁡x→∞f′(x)\lim \limits_{x \to \infty} f'(x) may not exist. [*] lim⁡x→∞f′(x)\lim \limits_{x \to \infty} f'(x) exists if ff is twice differentiable.
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