Let $f(x)$ be a function such that $\lim_{x \to \infty}f(x)=1$ and $\lim_{x \to \infty}f'(x)=\beta$. Then (1) $\beta > 1$, (2) $\beta$ must be $0$, (3) $\beta$ need not be $0$ but $|\beta| < 1$, (4) $\beta < -1$.
How I proceed: Take $f(x)=\frac{1}{x}$ then $f'(x)=-\frac{1}{x^2}$. Here $\beta=0$.So,(1) and (4) is not true.I am stuck between (2) and (3). Please help.