Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.3.44

Capitolo 2, Problema 2.3.44

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx → √3 1/x² = 1/3

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Understand the formal definition of a limit: For a function f(x) and a limit L, we say that lim(x → a) f(x) = L if for every ε > 0, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.

Identify the function and the limit in the problem: Here, f(x) = 1/x² and L = 1/3, with a = √3.

Set up the inequality based on the definition: We need to show that for every ε > 0, there exists a δ > 0 such that 0 < |x - √3| < δ implies |1/x² - 1/3| < ε.

Simplify the inequality |1/x² - 1/3| < ε: This can be rewritten as |(3 - x²)/(3x²)| < ε. Focus on finding a δ that satisfies this condition.

Choose δ appropriately: Analyze the expression |3 - x²| < 3εx² and find a δ that works for the given ε. Consider the behavior of the function around x = √3 to ensure the inequality holds.

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Limit Definition

The formal definition of a limit states that for a function f(x) to have a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.

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Definition of the Definite Integral

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity helps in evaluating limits, especially when dealing with rational functions like 1/x², as it ensures that the function behaves predictably near the limit point.

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Intro to Continuity

Rational Functions

Rational functions are ratios of polynomials, and their limits can often be evaluated by direct substitution unless the function is undefined at that point. In this case, analyzing the behavior of 1/x² as x approaches √3 is essential for determining the limit, which simplifies to 1/(√3)² = 1/3.

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Intro to Rational Functions

Pratica correlata

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Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2

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Limits and Infinity

Find the limits in Exercises 37–46.

2x² + 3

lim -------------

x→⁻∞ 5x² + 7

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