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

Verified step by step guidance

1. First, identify the form of the limit as \( x \to 0 \) for the expression \( \frac{x \sin(x^2)}{\tan^3 x} \). Substitute \( x = 0 \) to check if it results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
2. Since direct substitution gives \( 0 \) in the numerator (because \( x \to 0 \) and \( \sin(0) = 0 \)) and \( 0 \) in the denominator (because \( \tan(0) = 0 \)), the limit is of the indeterminate form \( \frac{0}{0} \), so l’Hôpital’s Rule applies.
3. 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 \( \tan^3 x \) using the chain rule: \( \frac{d}{dx}[\tan^3 x] = 3 \tan^2 x \cdot \sec^2 x \).
4. Rewrite the limit as \( \lim_{x \to 0} \frac{\sin(x^2) + 2x^2 \cos(x^2)}{3 \tan^2 x \sec^2 x} \) after differentiation.
5. 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.

Key Concepts

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Limit of Trigonometric Functions as x Approaches 0

Understanding the behavior of trigonometric functions near zero is crucial. For example, sin(x) ≈ x and tan(x) ≈ x when x is close to 0, which helps simplify expressions and evaluate limits involving these functions.

Chain Rule for Differentiation

The chain rule is used to differentiate composite functions, such as sin(x²) or tan³(x). It involves differentiating the outer function and multiplying by the derivative of the inner function, which is essential when applying l’Hôpital’s Rule to complex expressions.