Q22Multiple correct3 Marks7 Aug 2022

Choose the correct options from the following.

A

The function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = 2\sin(x) + 3\cos(x)$ is a bounded function.

B

Consider two real sequences $\{a_n\}$ and $\{b_n\}$ such that at least one of the limits $\lim_{n \to \infty} a_n$ or $\lim_{n \to \infty} b_n$ does not exist. Then the limit $\lim_{n \to \infty} (a_n + b_n)$ also does not exist.

C

The limit of the sequence $a_n = (-1)^n \cos(n\pi)$ is $1$.

D

If the derivatives of two functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are the same at all points, then $f(x) = g(x)$ for all $x \in \mathbb{R}$.