Textbook Question
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 2.6.62a
Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.
lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as
a. x → 0⁺
Verified step by step guidance1
Identify the individual terms in the expression: \( \frac{1}{x^{1/3}} \) and \( \frac{1}{(x-1)^{4/3}} \).
Consider the behavior of each term as \( x \to 0^+ \). For \( \frac{1}{x^{1/3}} \), as \( x \to 0^+ \), \( x^{1/3} \to 0^+ \), so \( \frac{1}{x^{1/3}} \to +\infty \).
For the term \( \frac{1}{(x-1)^{4/3}} \), as \( x \to 0^+ \), \( x-1 \to -1 \), so \( (x-1)^{4/3} \to 1 \) because the cube root of a negative number raised to the fourth power is positive. Thus, \( \frac{1}{(x-1)^{4/3}} \to 1 \).
Combine the limits of the two terms: \( \lim_{x \to 0^+} \left( \frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}} \right) = \lim_{x \to 0^+} \frac{1}{x^{1/3}} - \lim_{x \to 0^+} \frac{1}{(x-1)^{4/3}} \).
Since \( \frac{1}{x^{1/3}} \to +\infty \) and \( \frac{1}{(x-1)^{4/3}} \to 1 \), the overall limit is \( +\infty - 1 = +\infty \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function
The limit of a function describes the behavior of the function as the input approaches a particular value. In this context, we are interested in the limit as x approaches 0 from the positive side (x → 0⁺), which involves understanding how the function behaves near this point and whether it approaches a finite number, infinity, or negative infinity.
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Limits of Rational Functions: Denominator = 0
One-Sided Limits
One-sided limits consider the behavior of a function as the input approaches a specific value from one side only, either from the left (x → a⁻) or the right (x → a⁺). In this problem, x → 0⁺ indicates we are examining the limit as x approaches 0 from the positive side, which is crucial for understanding the function's behavior in this specific direction.
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Indeterminate Forms
Indeterminate forms occur in limits when the expression does not initially provide a clear answer, such as 0/0 or ∞ - ∞. These forms require further analysis or algebraic manipulation to resolve. In this problem, the expression involves terms that may lead to indeterminate forms as x approaches 0, necessitating techniques like simplification or substitution to find the limit.
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Related Practice
Textbook Question
Limits and Continuity
On what intervals are the following functions continuous?
a. ƒ(x) = x¹/³
Textbook Question
Limits and Continuity
On what intervals are the following functions continuous?
b. g(x) = x³/⁴
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let f(x) = (x² - 9) / (x + 3)
b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.
Textbook Question
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
b. limx→2 f(x) does not exist
Textbook Question
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(x)=x²−2x
a. [1, 3]