Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.102. lim(x→0) (x sin(x²))/(tan³x)

Accepted Answer

First, identify the form of the limit as $$ x 	o 0 $$ for the expression $$ \frac{x \sin(x^2)}{	an^3 x} . $$Substitute $$ x = 0  to $$check if it results in an indeterminate form like $$ \frac{0}{0}  or$$ $$ \frac{\infty}{\infty} . $$Since direct substitution gives $$ 0  in $$the numerator (because $$ x 	o 0 $$ and $$ \sin(0) = 0 ) $$and $$ 0  in $$the denominator (because $$ 	an(0) = 0 $$), 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 $$ x . $$For the numerator, use the product rule on $$ x \sin(x^2) $$: $$ \frac{d}{dx}[x \sin(x^2)] = \sin(x^2) + x \cdot \cos(x^2) \cdot 2x . $$For the denominator, differentiate $$ 	an^3 x $$ using the chain rule: $$ \frac{d}{dx}[	an^3 x] = 3 	an^2 x \cdot \sec^2 x . $$Rewrite the limit as $$ \lim_{x 	o 0} \frac{\sin(x^2) + 2x^2 \cos(x^2)}{3 	an^2 x \sec^2 x} $$ after differentiation. Evaluate the new limit by substituting $$ x = 0 $$ again. If the expression is still indeterminate, consider applying l’Hôpital’s Rule a second time or use series expansions to simplify the expression.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
102. lim(x→0) (x sin(x²))/(tan³x)

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Key Concepts

l’Hôpital’s Rule

Limit of Trigonometric Functions as x Approaches 0

Chain Rule for Differentiation