Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.7.37
Chapter 4, Problem 4.7.37

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))

Verified step by step guidance
1
Identify the form of the limit as x approaches π/2 from the left. Both the numerator tan(x) and the denominator 3/(2x - π) approach infinity, creating an indeterminate form of type ∞/∞.
Since the limit is in the indeterminate form ∞/∞, apply l'Hôpital's Rule, which allows us to differentiate the numerator and the denominator separately.
Differentiate the numerator: The derivative of tan(x) with respect to x is sec²(x).
Differentiate the denominator: The derivative of 3/(2x - π) with respect to x is -6/(2x - π)².
Re-evaluate the limit using the derivatives: lim_{x→π/2⁻} (sec²(x)) / (-6/(2x - π)²). Simplify the expression and evaluate the limit as x approaches π/2 from the left.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

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 infinity. Evaluating limits is crucial for defining derivatives and integrals, which are core components of calculus.
Recommended video:
05:50
One-Sided Limits

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 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. This rule simplifies the process of finding limits in complex functions.
Recommended video:
Guided course
5:50
Power Rules

Tangent Function

The tangent function, denoted as tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has vertical asymptotes where the cosine function is zero, which occurs at odd multiples of π/2. Understanding the behavior of the tangent function near these points is essential for evaluating limits involving tan(x).
Recommended video:
05:13
Slopes of Tangent Lines
Related Practice
Textbook Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

266
views
Textbook Question

Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. <IMAGE>

530
views
Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)

233
views
Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = (x+4)/(4-x)

261
views
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

219
views
Textbook Question

Sketch the graph of a function continuous on the given interval that satisfies the following conditions.

ƒ is continuous on the interval [-4, 4] ; f'(x) = 0 for x = -2, 0, and 3; ƒ has an absolute minimum at x = 3; ƒ has a local minimum at x = -2 ; ƒ has a local maximum at x = 0; ƒ has an absolute maximum at x = -4.

304
views