Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.R.81

Chapter 4, Problem 4.R.81

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx

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First, recognize that the limit is in an indeterminate form of 0^0 as x approaches 1. This suggests that we might need to use a logarithmic transformation to simplify the expression.

Take the natural logarithm of the expression: ln((x - 1)^sin(πx)) = sin(πx) * ln(x - 1). This allows us to transform the power into a product, which is easier to handle.

Now, evaluate the limit of the transformed expression: lim_(x→1) sin(πx) * ln(x - 1). Notice that as x approaches 1, sin(πx) approaches 0 and ln(x - 1) approaches -∞, creating an indeterminate form of 0 * -∞.

To resolve this, rewrite the expression as a quotient: lim_(x→1) (sin(πx)) / (1/ln(x - 1)). This is now in the form 0/0, which is suitable for l'Hôpital's Rule.

Apply l'Hôpital's Rule by differentiating the numerator and the denominator: differentiate sin(πx) with respect to x to get πcos(πx), and differentiate 1/ln(x - 1) with respect to x using the chain rule. Evaluate the new limit as x approaches 1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex scenarios.

Exponential Functions and Continuity

Exponential functions, such as those involving sin or cos, can exhibit unique behaviors near certain points. In the given limit, (x - 1) approaches 0 as x approaches 1, while sin(πx) oscillates between -1 and 1. Understanding the continuity of these functions and their limits is essential for accurately evaluating the limit, especially when combined with the properties of exponential growth or decay.

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Related Practice

Textbook Question

104–107. Functions from derivatives Find the function f with the following properties.

h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)

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Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x sin⁻¹ x on [-1, 1]

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Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_x→0 csc x sin⁻¹ x

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Textbook Question

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

a. What is the domain of f (in terms of a)?

421