Use l’Hôpital’s rule to find the limits sin x - xlim

We have to find the limit using L' Hospital's rule: limx→0sin x-xx3 We know in L' Hospital's rule we…

Answered: sin x - x lim

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.4: Values Of The Trigonometric Functions

Problem 17E

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Use l’Hôpital’s rule to find the limits

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Step 1

We have to find the limit using L' Hospital's rule:

$\lim_{x \to 0} \frac{\sin x - x}{x^{3}}$

We know in L' Hospital's rule we differentiate numerator as well as denominator

so differentiating (since limit is $\frac{0}{0}$ form)

$\begin{array}{rcl} \lim_{x \to 0} \frac{\sin x - x}{x^{3}} & = & \lim_{x \to 0} \frac{\frac{d (\sin x - x)}{d x}}{\frac{d (x^{3})}{d x}} \\ = & \lim_{x \to 0} \frac{\frac{d \sin x}{d x} - \frac{d x}{d x}}{3 x^{3 - 1}} \\ = & \lim_{x \to 0} \frac{\cos x - 1}{3 x^{2}} \end{array}$

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