Types of Infinity — Question 1

Question 1

Prove that $$\lim_{x\to\infty}\frac{\ln x}{x}=0.$$

Question 1 - Solution

We compare the growth rates of the numerator and denominator.

For $x>0$, both $\ln x$ and $x$ are differentiable. Apply L’Hôpital’s Rule to the limit $$\lim_{x\to\infty}\frac{\ln x}{x}.$$

Differentiate the numerator and denominator: $$\frac{d}{dx}(\ln x)=\frac{1}{x}, \qquad \frac{d}{dx}(x)=1.$$

Thus, $$\lim_{x\to\infty}\frac{\ln x}{x} = \lim_{x\to\infty}\frac{1/x}{1} = \lim_{x\to\infty}\frac{1}{x}.$$

From basic limits, $$\lim_{x\to\infty}\frac{1}{x}=0.$$

Therefore, $$\lim_{x\to\infty}\frac{\ln x}{x}=0.$$

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