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 → −∞ (2x + √(4x² + 3x − 2))
1. Limits and Continuity
Introduction to Limits
3:55 minutesProblem 2.6.95
Textbook Question
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 3 (−2 / (x − 3)²) = −∞
Verified step by step guidance1
Understand the problem: We need to prove that the limit of the function \(-\frac{2}{(x-3)^2}\) as \(x\) approaches 3 is \(-\infty\). This involves using the formal definition of limits involving infinity.
Recall the formal definition: For a limit to be \(-\infty\) as \(x\) approaches a value \(c\), for every positive number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(f(x) < -M\).
Apply the definition: We need to show that for every \(M > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 3| < \delta\), then \(-\frac{2}{(x-3)^2} < -M\).
Solve the inequality: Start by solving \(-\frac{2}{(x-3)^2} < -M\). This simplifies to \(\frac{2}{(x-3)^2} > M\). Rearrange to find \(x\) values that satisfy this inequality.
Determine \(\delta\): From the inequality \((x-3)^2 < \frac{2}{M}\), solve for \(x\) to find \(\delta\) such that \(0 < |x - 3| < \delta\) ensures the inequality holds. This will complete the proof using the formal definition.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
The formal definition of a limit involves showing that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In this context, proving a limit involves demonstrating that as x approaches a specific value, the function approaches a particular limit, which can be finite or infinite.
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Infinite Limits
Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. In this problem, the limit is negative infinity, indicating that as x approaches 3, the function value decreases indefinitely. Understanding infinite limits requires recognizing how the function behaves near the point of interest.
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Behavior Near Singularities
Singularities are points where a function is not defined or behaves erratically, often leading to infinite limits. In this problem, x = 3 is a singularity for the function −2/(x−3)², as the denominator approaches zero, causing the function to diverge. Analyzing behavior near singularities involves examining how the function's value changes as it approaches these critical points.
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