How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$ [duplicate]
(a) comes from the Mean Value Theorem. For $\varepsilon>0$ there is $x_0$ such that $L-\varepsilon<f'(x)<L+\varepsilon$ for all $x\ge x_0$. If $x\ge x_0$ then $(f(x+h)-f(x))/h=f'(x+\xi)$ where $0<\xi<h$. Then $L-\varepsilon<f'(x+\xi)<L+\varepsilon$ etc.