Use L’Hôpital’s Rule to evaluate the limit. sin xlim* o sin 5 x

We have to find the limit using L' Hospital's rule: limx→0sin xsin 5x Since limit is in 00…

Answered: sin x lim * o sin 5 x

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.2: Trigonometric Equations

Problem 38E

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Question

Use L’Hôpital’s Rule to evaluate the limit.

Expert Solution

Step 1

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

$\lim_{x \to 0} \frac{\sin x}{\sin 5 x}$

Since limit is in $\frac{0}{0}$ indeterminate form so we can apply L' Hospital's rule.

According to this rule we need to differentiate the numerator and denominator till we find the limit in determinate form.

Applying L' Hospital's rule for the given limit, we get

$\begin{array}{rcl} \lim_{x \to 0} \frac{\frac{d (\sin x)}{d x}}{\frac{d (\sin 5 x)}{d x}} & = & \lim_{x \to 0} \frac{\cos x}{\cos 5 x (\frac{d 5 x}{d x})} \\ = & \lim_{x \to 0} \frac{\cos x}{\cos 5 x (5 \frac{d x}{d x})} \\ = & \lim_{x \to 0} \frac{\cos x}{\cos 5 x (5 \times 1)} \\ = & \lim_{x \to 0} \frac{\cos x}{5 \cos 5 x} \end{array}$

We have used following formula of differentiation,

$\begin{array}{rcl} \frac{d x^{n}}{d x} & = & n x^{n - 1} \\ \frac{d x}{d x} & = & 1 \\ \frac{d \sin x}{d x} & = & \cos x \\ \frac{d \sin f (x)}{d x} & = & \cos f (x) \frac{d f (x)}{d x} \end{array}$

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