Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.91. lim(x→π/2⁻) (sec(7x))(cos(3x))

Accepted Answer

First, rewrite the limit expression to understand its behavior as $$x$$ approaches $$\frac{\pi}{2}$$ from the left: $$\lim_{x 	o \frac{\pi}{2}^-} (\sec(7x)) (\cos(3x)). $$Recall that $$\sec(	heta) = \frac{1}{\cos(	heta)}$$, so rewrite the expression as $$\lim_{x 	o \frac{\pi}{2}^-} \frac{\cos(3x)}{\cos(7x)}. $$Evaluate the numerator and denominator separately at $$x = \frac{\pi}{2} to $$check if the limit is an indeterminate form: $$\cos(3 \cdot \frac{\pi}{2})$$ and $$\cos(7 \cdot \frac{\pi}{2}). If $$direct substitution results in an indeterminate form like $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, apply l’Hôpital’s Rule by differentiating the numerator and denominator with respect to $$x$$: $$\lim_{x 	o \frac{\pi}{2}^-} \frac{\frac{d}{dx} \cos(3x)}{\frac{d}{dx} \cos(7x)} = \lim_{x 	o \frac{\pi}{2}^-} \frac{-3 \sin(3x)}{-7 \sin(7x)} = \lim_{x 	o \frac{\pi}{2}^-} \frac{3 \sin(3x)}{7 \sin(7x)}.$$ Finally, evaluate this new limit by substituting $$x = \frac{\pi}{2}$$ and simplify to find the limit.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))

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