Use l’Hôpital’s rule to find the limits in Exercises 7–52.


22. lim (x → 1) (x - 1) / (ln x - sin πx)

Verified step by step guidance

1. First, verify that the limit is an indeterminate form of type \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) by substituting \( x = 1 \) into the expression \( \frac{x - 1}{\ln x - \sin \pi x} \).
2. Since direct substitution gives \( \frac{0}{0} \), apply l'Hôpital's Rule, which states that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \) if the original limit is an indeterminate form.
3. Find the derivative of the numerator: \( f(x) = x - 1 \), so \( f'(x) = 1 \).
4. Find the derivative of the denominator: \( g(x) = \ln x - \sin \pi x \). Use the derivatives \( \frac{d}{dx} \ln x = \frac{1}{x} \) and \( \frac{d}{dx} \sin \pi x = \pi \cos \pi x \), so \( g'(x) = \frac{1}{x} - \pi \cos \pi x \).
5. Evaluate the new limit \( \lim_{x \to 1} \frac{f'(x)}{g'(x)} = \lim_{x \to 1} \frac{1}{\frac{1}{x} - \pi \cos \pi x} \) by substituting \( x = 1 \).

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.

Limits Involving Logarithmic and Trigonometric Functions

Understanding how logarithmic functions (like ln x) and trigonometric functions (like sin πx) behave near specific points is crucial. This helps in simplifying expressions and determining if direct substitution leads to indeterminate forms.

Derivative Computation

Calculating derivatives of functions such as (x - 1), ln x, and sin πx accurately is essential when applying l’Hôpital’s Rule. Knowing the derivative rules for polynomials, logarithms, and trigonometric functions ensures correct application of the rule.