Ch. 7 - Transcendental Functions

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Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.5.24

Capitolo 7, Problema 7.5.24

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

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First, identify the form of the limit as \(x\) approaches \(\frac{\pi}{2}\). Evaluate the numerator and denominator separately at \(x = \frac{\pi}{2}\) to check if it results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

Rewrite the limit expression: \(\lim_{x \to \frac{\pi}{2}} \frac{\ln(\csc x)}{\left(x - \frac{\pi}{2}\right)^2}\). Note that \(\csc x = \frac{1}{\sin x}\), so \(\ln(\csc x) = \ln\left(\frac{1}{\sin x}\right) = -\ln(\sin x)\).

Since the denominator is squared, the limit is of the form \(\frac{0}{0}\) after substitution, so l’Hôpital’s Rule applies. Differentiate the numerator and denominator with respect to \(x\):

Calculate the derivative of the numerator: \(\frac{d}{dx} \left( \ln(\csc x) \right) = \frac{d}{dx} \left( -\ln(\sin x) \right) = -\frac{\cos x}{\sin x} = -\cot x\).

Calculate the derivative of the denominator: \(\frac{d}{dx} \left( x - \frac{\pi}{2} \right)^2 = 2 \left( x - \frac{\pi}{2} \right)\). Then, rewrite the limit as \(\lim_{x \to \frac{\pi}{2}} \frac{-\cot x}{2 \left( x - \frac{\pi}{2} \right)}\). If this still results in an indeterminate form, apply l’Hôpital’s Rule again by differentiating numerator and denominator once more.

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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. This rule simplifies complex limit problems by transforming them into easier derivative calculations.

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Power Rules

Behavior of Trigonometric Functions Near Specific Points

Understanding how trigonometric functions like cosecant (csc x) behave near points such as x = π/2 is crucial. Since csc x = 1/sin x, and sin(π/2) = 1, csc x is well-defined near π/2. Analyzing the limit involves knowing the continuity and differentiability of these functions around the point of interest.

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Introduction to Trigonometric Functions

Logarithmic Functions and Their Limits

The natural logarithm function, ln(x), is continuous and differentiable for x > 0. When combined with trigonometric functions inside the logarithm, it is important to ensure the argument remains positive near the limit point. Understanding how ln(csc x) behaves near π/2 helps in applying l’Hôpital’s Rule correctly.

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Graphs of Logarithmic Functions

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