Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - 12.7 L'Hospital's Rule - 12.7 Exercises - Page 660: 47

Answer

The l'Hospital's conditions are not fulfilled

Work Step by Step

The l'Hospital's rule is used when the limit has an indeterminte form type $\frac{0}{0}$ or $\frac{\infty}{\infty}$ The given limit is: $$\lim\limits_{x \to 0}\frac{x^{2}}{x^{2}+3}$$ So using the direct substitution of limit property since the function inside the limit is continuous at $x=0$: $$\lim\limits_{x \to 0}\frac{x^{2}}{x^{2}+3}=\lim\limits_{x \to 0}\frac{0^{2}}{0^{2}+3}=\lim\limits_{x \to 0}\frac{0}{3}=0$$ Since the limit does not have an indeterminate form type $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and exactly is equal to $0$ by the direct substitution of limit property so the l'Hospital's should not be used.
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