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 4.7.28
Chapter 4, Problem 4.7.28

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


lim_x→ 0⁺ (x - 3 √x) / (x - √x)

Verified step by step guidance
1
Identify the form of the limit as x approaches 0 from the positive side. Substitute x = 0 into the expression (x - 3√x) / (x - √x) to check if it results in an indeterminate form like 0/0.
Since substituting x = 0 gives 0/0, l'Hôpital's Rule can be applied. This rule states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, then the limit can be found by differentiating the numerator and the denominator separately.
Differentiate the numerator: The numerator is x - 3√x. The derivative of x is 1, and the derivative of 3√x is (3/2)x^(-1/2). Therefore, the derivative of the numerator is 1 - (3/2)x^(-1/2).
Differentiate the denominator: The denominator is x - √x. The derivative of x is 1, and the derivative of √x is (1/2)x^(-1/2). Therefore, the derivative of the denominator is 1 - (1/2)x^(-1/2).
Apply l'Hôpital's Rule: Take the limit of the new expression formed by the derivatives of the numerator and the denominator as x approaches 0 from the positive side. Evaluate lim_x→0⁺ [(1 - (3/2)x^(-1/2)) / (1 - (1/2)x^(-1/2))].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
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 where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
Recommended video:
05:50
One-Sided Limits

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) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, then re-evaluating the limit. This technique simplifies the process of finding limits in complex functions.
Recommended video:
Guided course
5:50
Power Rules

Square Roots and Their Properties

Square roots are mathematical functions that return the non-negative value whose square equals the given number. Understanding how to manipulate square roots is essential in calculus, especially when evaluating limits involving expressions with square roots. Simplifying expressions with square roots can often help in resolving indeterminate forms and making limits easier to compute.
Recommended video:
06:21
Properties of Functions
Related Practice
Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = sin 3x on [-π/4,π/3]

Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = x² - 10 on [-2, 3]

Textbook Question

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

lim_x→∞ (log₂ x - log₃ x)

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) = 3x³ - 4x

Textbook Question

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


lim_x→∞ (tan⁻¹ x - π/2)/(1/x)

Textbook Question

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


lim_z→0 (tan 4z) / (tan 7z)