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)
1. Limits and Continuity
Introduction to Limits
4:06 minutesProblem 100
Textbook Question
Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.
lim x→2⁻ (1 / (x − 2)) = −∞
Verified step by step guidance1
Understand the problem: We need to prove that as x approaches 2 from the left (x → 2⁻), the function f(x) = 1 / (x - 2) approaches negative infinity.
Recall the formal definition of a limit approaching negative infinity: For every positive number M, there exists a δ > 0 such that if 0 < |x - 2| < δ and x < 2, then f(x) < -M.
Set up the inequality: We want to show that 1 / (x - 2) < -M. This implies that x - 2 < -1/M, which further implies x < 2 - 1/M.
Choose δ: To satisfy the condition 0 < |x - 2| < δ and x < 2, we can choose δ = min(1, 1/M). This ensures that x is close to 2 from the left and satisfies the inequality.
Conclude the proof: With this choice of δ, whenever 0 < |x - 2| < δ and x < 2, the inequality 1 / (x - 2) < -M holds, thus proving that the limit of f(x) as x approaches 2 from the left is negative infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
The limit of a function at a point describes the behavior of the function as it approaches that point. Formally, we say that the limit of f(x) as x approaches a is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for proving limit statements rigorously.
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One-Sided Limits
One-sided limits consider the behavior of a function as it approaches a specific point from one side only. The left-hand limit, denoted as lim x→a⁻ f(x), examines the values of f(x) as x approaches a from the left. Understanding one-sided limits is essential for analyzing functions that exhibit different behaviors from either side of a point, particularly in cases of discontinuity.
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Behavior of Rational Functions
Rational functions are ratios of polynomials, and their limits can often lead to infinite values. In the case of lim x→2⁻ (1 / (x - 2)), as x approaches 2 from the left, the denominator approaches zero negatively, causing the function to decrease without bound. Recognizing how rational functions behave near their vertical asymptotes is key to understanding limits that result in ±∞.
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