Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

Accepted Answer

First, identify the form of the limit as $$x$$ approaches $$\frac{\pi}{2}. $$Evaluate the numerator and denominator separately at $$x = \frac{\pi}{2} to $$check if it results in an indeterminate form like $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}. $$Rewrite the limit expression: $$\lim_{x 	o \frac{\pi}{2}} \frac{\ln(\csc x)}{\left(x - \frac{\pi}{2}\$$right)^2$$}. $$Note that $$\csc x = \frac{1}{\sin x}$$, so $$\ln(\csc x) = \ln\left(\frac{1}{\sin x}ight) = -\ln(\sin x). $$Since the denominator is squared, the limit is of the form $$\frac{0}{0}$$ after substitution, so l’Hôpital’s Rule applies. Differentiate the numerator and denominator with respect to $$x$$: Calculate the derivative of the numerator: $$\frac{d}{dx} \left( \ln(\csc x) ight) = \frac{d}{dx} \left( -\ln(\sin x) ight) = -\frac{\cos x}{\sin x} = -\cot x. $$Calculate the derivative of the denominator: $$\frac{d}{dx} \left( x - \frac{\pi}{2} \$$right)^2$$ = 2 \left( x - \frac{\pi}{2} ight). $$Then, rewrite the limit as $$\lim_{x 	o \frac{\pi}{2}} \frac{-\cot x}{2 \left( x - \frac{\pi}{2} ight)}. If $$this still results in an indeterminate form, apply l’Hôpital’s Rule again by differentiating numerator and denominator once more.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

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Key Concepts

l’Hôpital’s Rule

Behavior of Trigonometric Functions Near Specific Points

Logarithmic Functions and Their Limits