Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 311: 21


$$\lim_{x\to0^+}\frac{\ln x}{x}=-\infty$$

Work Step by Step

$$A=\lim_{x\to0^+}\frac{\ln x}{x}$$ As $\lim_{x\to0^+}(\ln x)=-\infty$ (as $x\to0^+$, $\ln x\to-\infty$) and $\lim_{x\to0^+}x=0$ this is a form of $\frac{-\infty}{0}$, so L'Hospital's Rule cannot be applied. However, we can still judge intuitively. As $x\to0^+$, $\ln x\to-\infty$, a negatively very large number, while $x\to0^+$, a very small positive number (but never reaches $0$). So we see here that the limit of the numerator has already reached negative infinity. When being divided by a very small positive number, it would become even smaller, going further into negative infinity. In other words, $$A=\lim_{x\to0^+}\frac{\ln x}{x}=-\infty.$$
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