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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.5.22
Capitolo 7, Problema 7.5.22

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


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

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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 \( \frac{x - 1}{\ln x - \sin \pi x} \).
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.
Find the derivative of the numerator: \( f(x) = x - 1 \), so \( f'(x) = 1 \).
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 \( g'(x) = \frac{1}{x} - \pi \cos \pi x \).
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 \( x = 1 \).

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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.
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Power Rules

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.
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Introduction to Trigonometric Functions

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.
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Derivatives
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