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IMC 1995 Problem 3

Source: IMC 1995

February 18, 2021
differentiable functionreal analysis

Problem Statement

Let ff be twice continuously differentiable on (0,∞)(0,\infty) such that lim⁡x→0+f′(x)=−∞\lim_{x \to 0^{+}}f'(x)=-\infty and lim⁡x→0+f′′(x)=∞\lim_{x \to 0^{+}}f''(x)=\infty. Show that lim⁡x→0+f(x)f′(x)=0.\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.
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