Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.R.75

Chapter 4, Problem 4.R.75

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→π / 2- (sin x) ^tan x

Verified step by step guidance

Step 1: Identify the form of the limit as x approaches π/2 from the left. The expression (sin x)^tan x can be rewritten using the exponential function: e^(tan x * ln(sin x)).

Step 2: Analyze the behavior of sin x and tan x as x approaches π/2 from the left. Note that sin x approaches 1 and tan x approaches infinity.

Step 3: Recognize that the limit involves an indeterminate form of type 1^∞. To resolve this, take the natural logarithm of the expression, which transforms the problem into finding the limit of tan x * ln(sin x) as x approaches π/2 from the left.

Step 4: Apply l'Hôpital's Rule to the limit of tan x * ln(sin x). This requires differentiating the numerator and the denominator separately. The derivative of tan x is sec^2 x, and the derivative of ln(sin x) is cot x.

Step 5: Evaluate the new limit using l'Hôpital's Rule. If necessary, apply l'Hôpital's Rule multiple times until the limit can be determined. Once the limit of the logarithmic expression is found, exponentiate the result to find the original limit.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches π/2 involves analyzing the behavior of the function near that point, which may lead to indeterminate forms.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits when direct substitution is not possible.

Trigonometric Functions

Trigonometric functions, such as sine and tangent, are periodic functions that relate angles to ratios of sides in right triangles. Understanding their behavior, especially near critical points like π/2, is crucial for limit evaluation. In this problem, the sine function approaches 1 as x approaches π/2, while the behavior of tan x must also be considered to determine the overall limit.

Recommended video:

6:04

Introduction to Trigonometric Functions

Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.

views

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).

217

views

Textbook Question

Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

275

views

Textbook Question

Maximum area A line segment of length 10 joins the points (0, p) and (q, 0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.

180

views

Textbook Question

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

lim_ x→0 ⁺ | ln x | ˣ

224

views

Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

268

views