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f(x)/x=∞ for x->∞, integral is convergent of sin(f)

Source: VJIMC 2006 2.4

June 29, 2021
calculusintegration

Problem Statement

Let f:[0,∞)→Rf:[0,\infty)\to\mathbb R ba a strictly convex continuous function such that lim⁡x→+∞f(x)x=+∞.\lim_{x\to+\infty}\frac{f(x)}x=+\infty.Prove that the improper integral ∫0+∞sin⁡(f(x))dx\int^{+\infty}_0\sin(f(x))\text dx is convergent but not absolutely convergent.
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