Use L'Hôpital's Rule (possibly more than once) to evaluate the following limitsin(öz)-őz cos(6z)ör- sin(6z)lim%3DIf the answer equals o or -o, write INF or -INF in the blank.

Given, limx→0sin6x-6xcos6x6x-sin6x

Answered: Use L'Hôpital's Rule (possibly more…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.3: Trigonometric Functions Of Real Numbers

Problem 32E

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Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit
sin(öz)-őz cos(6z)
ör- sin(6z)
lim
%3D
If the answer equals o or -o, write INF or -INF in the blank.

Expert Solution

Step 1

$G i v e n, \lim_{x \to 0} (\frac{\sin (6 x) - 6 x \cos (6 x)}{6 x - \sin (6 x)})$

Step 2

$N o w, ∴ \lim_{x \to 0} (\frac{\sin (6 x) - 6 x \cos (6 x)}{6 x - \sin (6 x)}) A p p l y i n g L' H o s p i t a l' s r u l e, w e h a v e = \lim_{x \to 0} (\frac{6 \cos (6 x) - 6 [x (- 6 \sin (6 x)) + \cos (6 x)]}{6 - 6 \cos (6 x)}) = \lim_{x \to 0} (\frac{6 \cos (6 x) + 36 x \sin (6 x) - 6 \cos (6 x)}{6 - 6 \cos (6 x)}) = \lim_{x \to 0} (\frac{6 x \sin (6 x)}{1 - \cos (6 x)}) A g a i n, a p p l y i n g L' H o s p i t a l' s r u l e, w e h a v e = \lim_{x \to 0} (\frac{6 [6 x \cos (6 x) + \sin (6 x)]}{0 - (- \sin (6 x)) 6}) = \lim_{x \to 0} (\frac{6 x \cos (6 x) + \sin (6 x)}{\sin (6 x)})$

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