L'Hopital's Rule Math Example 4

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Example 4

medium
Find lim⁡x→∞ln⁡xx\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}.

Solution

  1. 1
    ∞∞\frac{\infty}{\infty} form. Apply L'Hôpital: lim⁡x→∞1/x1=lim⁡x→∞1x=0\lim_{x\to\infty}\frac{1/x}{1} = \lim_{x\to\infty}\frac{1}{x} = 0.

Answer

00
ln⁡x\ln x grows to infinity but far slower than xx. L'Hôpital confirms this: after one application the limit is 1/x→01/x \to 0.

About L'Hopital's Rule

If lim⁡x→af(x)g(x)\lim_{x \to a} \frac{f(x)}{g(x)} is an indeterminate form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, then lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} provided the right-hand limit exists (or is ±∞\pm\infty).

Learn more about L'Hopital's Rule →

More L'Hopital's Rule Examples

Example 1 easy

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Example 2 hard

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Example 3 easy

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