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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 2.6.95
Capitolo 2, Problema 2.6.95

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


lim x → 3 (−2 / (x − 3)²) = −∞

Guida verificata passo dopo passo
1
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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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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Percorso guidato
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Definition of the Definite Integral

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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One-Sided Limits

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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Cases Where Limits Do Not Exist
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Using Limit Rules


Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.


limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)


(We assume the denominator is nonzero.)


(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)


= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)


= ((2)(1) − (−5)) / (1 + 7)² = 7/64

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