Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

Accepted Answer

First, identify the form of the limit as $$ 	heta 	o \frac{\pi}{2} . $$Substitute $$ 	heta = \frac{\pi}{2} $$ into the expression $$ \frac{2	heta - \pi}{\cos(2\pi - 	heta)}  to $$check if it results in an indeterminate form like $$ \frac{0}{0}  or$$ $$ \frac{\infty}{\infty} . $$Since direct substitution gives $$ 2\left(\frac{\pi}{2}ight) - \pi = 0 $$ and $$ \cos(2\pi - \frac{\pi}{2}) = \cos\left(\frac{3\pi}{2}ight) = 0 $$, the limit is of the indeterminate form $$ \frac{0}{0} $$, so l’Hôpital’s Rule applies. Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to $$ 	heta . $$The derivative of the numerator $$ 2	heta - \pi  is$$ $$ 2 . $$The derivative of the denominator $$ \cos(2\pi - 	heta) $$ requires the chain rule: $$ \frac{d}{d	heta} \cos(2\pi - 	heta) = -\sin(2\pi - 	heta) 	imes (-1) = \sin(2\pi - 	heta) . $$Rewrite the limit as $$ \lim_{	heta 	o \frac{\pi}{2}} \frac{2}{\sin(2\pi - 	heta)} $$ and then substitute $$ 	heta = \frac{\pi}{2}  to $$evaluate the limit.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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Key Concepts

Limits and Indeterminate Forms

l’Hôpital’s Rule

Trigonometric Functions and Their Limits