Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 3 (−2 / (x − 3)²) = −∞

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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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주요 개념

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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.

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.

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.

Use formal definitions to prove the limit statements in Exercises 93–96.lim x → 3 (−2 / (x − 3)²) = −∞