Assume that f and g are differentiable functions defined on all of the real line.If f ' > 0 everywhere and f > 0 everywhere then must $\operatorname{lim}_{x \rightarrow+\infty} f(x)=\infty$ ?

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Quizplus · Accepted Answer

The fact that $f' > 0$ everywhere means $f$ is increasing everywhere, and $f > 0$ everywhere means $f$ is positive. However, this does not guarantee that $f(x)$ approaches infinity as $x$ approaches infinity. $f(x)$ could approach a finite limit or increase indefinitely without bound, but more slowly than approaching infinity.

Assume That F and G Are Differentiable Functions Defined on All