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

Verified step by step guidance

1. First, identify the form of the limit as \( x \to 0 \) for the expression \( \frac{x - \sin x}{x \tan x} \). 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 \( \frac{0 - 0}{0 \cdot 0} = \frac{0}{0} \), 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 \( x \to a \) is \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then the limit equals \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided this latter limit exists.
3. Compute the derivative of the numerator: \( f(x) = x - \sin x \). Using basic differentiation rules, \( f'(x) = 1 - \cos x \).
4. Compute the derivative of the denominator: \( g(x) = x \tan x \). Use the product rule: \( g'(x) = \frac{d}{dx}(x) \cdot \tan x + x \cdot \frac{d}{dx}(\tan x) = 1 \cdot \tan x + x \cdot \sec^2 x \).
5. Rewrite the limit using these derivatives: \[ \lim_{x \to 0} \frac{1 - \cos x}{\tan x + x \sec^2 x} \]. Then, evaluate this new limit by substituting \( x = 0 \) or apply l’Hôpital’s Rule again if necessary.

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.

Trigonometric Limits and Functions

Understanding the behavior of trigonometric functions such as sin x and tan x near zero is essential. Knowing standard limits like lim (x→0) (sin x)/x = 1 helps simplify expressions and apply l’Hôpital’s Rule effectively.

Derivative of Trigonometric Functions

Calculating derivatives of sin x and tan x accurately is crucial when applying l’Hôpital’s Rule. For example, the derivative of sin x is cos x, and the derivative of tan x is sec² x, which are used to find the new limit after differentiation.