TIFR Mathematics 2022 | Part B | Question: 17

Question

Answer whether the following statements are True or False.

There exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
\[ \lim _{x \rightarrow \infty} f(x)=2 \quad \text { and } \quad \lim _{x \rightarrow \infty} f^{\prime}(x)=1 .\]

Answer 1

False. Because if f(x) is 2 when x-->infinity, then it means that when x-->infinity then f(x) is constant and we know that slope of a constant function is 0. So, when x-->infinity, then f’(x) i.e.,slope is 0.

Mathematically,