Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.49. lim (x → 0) (x - sin x) / (x tan x)

Accepted Answer

First, identify the form of the limit as $$ x 	o 0 $$ for the expression $$ \frac{x - \sin x}{x 	an x} . $$Substitute $$ x = 0  to $$check if it results in an indeterminate form like $$ \frac{0}{0}  or$$ $$ \frac{\infty}{\infty} . $$Since direct substitution gives $$ \frac{0 - 0}{0 \cdot 0} = \frac{0}{0} $$, which is an indeterminate form, we can apply l’Hôpital’s Rule. This rule states that if the limit of $$ \frac{f(x)}{g(x)}  as$$ $$ x 	o a  is$$ $$ \frac{0}{0}  or$$ $$ \frac{\infty}{\infty} $$, then the limit equals $$ \lim_{x 	o a} \frac{f'(x)}{g'(x)} $$, provided this latter limit exists. Compute the derivative of the numerator: $$ f(x) = x - \sin x . $$Using basic differentiation rules, $$ f'(x) = 1 - \cos x . $$Compute the derivative of the denominator: $$ g(x) = x 	an x . $$Use the product rule: $$ g'(x) = \frac{d}{dx}(x) \cdot 	an x + x \cdot \frac{d}{dx}(	an x) = 1 \cdot 	an x + x \cdot \sec^2 x . $$Rewrite the limit using these derivatives: $$ \lim_{x 	o 0} \frac{1 - \cos x}{	an x + x \sec^2 x} . $$Then, evaluate this new limit by substituting $$ x = 0  or $$apply l’Hôpital’s Rule again if necessary.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)

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Key Concepts

l’Hôpital’s Rule

Trigonometric Limits and Functions

Derivative of Trigonometric Functions