Textbook Question
Limits of Rational Functions
In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.
f(x) = (x + 1)/(x² + 3)
365
views
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 2.3.41
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→1 f(x) = 1 if f(x) = {x², x ≠ 1
2, x = 1
Verified step by step guidance1
Understand the formal definition of a limit: For the limit \( \lim_{x \to a} f(x) = L \), for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
Identify the function and the limit: Here, \( f(x) = x^2 \) when \( x \neq 1 \) and \( f(x) = 2 \) when \( x = 1 \). We need to prove that \( \lim_{x \to 1} f(x) = 1 \).
Set \( L = 1 \) and \( a = 1 \). We need to show that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 1| < \delta \), then \( |x^2 - 1| < \epsilon \).
Simplify the expression \( |x^2 - 1| \): Notice that \( |x^2 - 1| = |(x - 1)(x + 1)| \). We need to find \( \delta \) such that \( |(x - 1)(x + 1)| < \epsilon \).
Choose \( \delta \) such that \( |x - 1| < \delta \) and ensure that \( |x + 1| \) is bounded. For instance, if \( |x - 1| < 1 \), then \( 0 < x < 2 \), so \( |x + 1| < 3 \). Then, choose \( \delta = \min(1, \frac{\epsilon}{3}) \) to satisfy the condition \( |(x - 1)(x + 1)| < \epsilon \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
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 as x approaches a particular value is the value that the function approaches as the input gets arbitrarily close to that point. In this context, limx→1 f(x) = 1 means that as x gets closer to 1, f(x) approaches 1, regardless of the function's value at x = 1.
Recommended video:
06:11
Limits of Rational Functions: Denominator = 0
Formal Definition of a Limit
The formal definition of a limit, often called the epsilon-delta definition, states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is used to rigorously prove that the limit of f(x) as x approaches c is L, ensuring the function's behavior near the point is consistent with the limit value.
Recommended video:
05:43Definition of the Definite Integral
Piecewise Functions
A piecewise function is defined by different expressions based on the input value. In this problem, f(x) is x² when x ≠ 1 and 2 when x = 1. Understanding how to handle piecewise functions is crucial for evaluating limits, as the limit depends on the behavior of the function as x approaches the limit point, not necessarily the function's value at that point.
Recommended video:
05:36Piecewise Functions
Related Practice
Textbook Question
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x + 9) − √(x + 4))
325
views
Textbook Question
Calculating Limits
Find the limits in Exercises 11–22.
limx→−1/2 4x(3x+4)²
218
views
Textbook Question
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 15x + 1 = 0 (three roots)
338
views
Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x
236
views
Textbook Question
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
f(x) = 2/x − 3
314
views