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 4.7.75
Chapter 4, Problem 4.7.75

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


lim_x→0⁺ x²ˣ

Verified step by step guidance
1
Identify the form of the limit as x approaches 0 from the positive side. The expression x²ˣ can be rewritten as e^(x ln(x²)). As x approaches 0⁺, x ln(x²) approaches an indeterminate form 0 * (-∞).
Rewrite the expression x²ˣ as e^(2x ln(x)). This helps in applying the limit to the exponent first, which is 2x ln(x).
Consider the limit of the exponent: lim_(x→0⁺) 2x ln(x). This is an indeterminate form of type 0 * (-∞), which can be rewritten as lim_(x→0⁺) (ln(x) / (1/(2x))).
Apply l'Hôpital's Rule to the limit of the exponent, since it is in the indeterminate form ∞/∞. Differentiate the numerator and the denominator separately: the derivative of ln(x) is 1/x, and the derivative of 1/(2x) is -1/(2x²).
Evaluate the new limit after applying l'Hôpital's Rule: lim_(x→0⁺) (1/x) / (-1/(2x²)) simplifies to lim_(x→0⁺) -2x. As x approaches 0⁺, this limit approaches 0. Therefore, the original limit of x²ˣ is e^0, which simplifies to 1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
11m
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 essential for defining derivatives and integrals, which are core components of calculus.
Recommended video:
05:50
One-Sided Limits

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. In the context of the limit lim_x→0⁺ x²ˣ, the expression involves an exponential function with a variable base. Understanding the properties of exponential functions, especially their behavior as the exponent approaches zero, is crucial for evaluating this limit.
Recommended video:
6:13
Exponential Functions

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 ∞/∞. 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. This rule simplifies the process of finding limits, especially when direct substitution is not possible.
Recommended video:
Guided course
5:50
Power Rules
Related Practice
Textbook Question

Explain the Mean Value Theorem with a sketch.

Textbook Question

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.


f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0

Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x³ - 3x² + 12

Textbook Question

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


lim_u→ π/4 (tan u - cot u) / (u - π/4)

Textbook Question

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

1
views
Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = x² √(x + 5)

1
views