Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.7 L'Hopital's Rule - 4.7 Exercises: 1

Answer

It is meant that both the denominator and the numerator of the expression under the limit approach $0$ as $x$ tends to $a$. The examples are $$\lim_{x\to0}\frac{\sin x}{x},\quad \lim_{x\to0}\frac{x^3}{2x},\quad \lim_{x\to0}\frac{1-\cos x}{\sin x}.$$

Work Step by Step

It is meant that when $x\to a$ then both the numerator and the denominator of the expression under the limit approach zero. The examples are 1) $$\lim_{x\to 0}\frac{\sin x}{x}$$ because as $x\to0$ the numerator $\sin x$ approaches $0$ (sine is a continuous function and $\sin 0 =0$) and the denominator is just $x$ that already goes to zero. 2) $$\lim_{x\to0}\frac{x^3}{2x}.$$ When $x\to0$ then every positive power of $x$ approaches zero so both $x^3$ and $2x$ tend to zero. 3) $$\lim_{x\to0}\frac{1-\cos x}{\sin x}.$$ We already showed that when $x\to0 $ then $\sin x\to0$. Also when $x\to 0$ then $\cos x\to 1$ because cosine is a continuous fucntion and $\cos 0=1$. Thus $1-\cos x$ tends to zero.
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