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 2.6.37

Capitolo 2, Problema 2.6.37

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0⁺ 1 / 3x

Guida verificata passo dopo passo

Step 1: Understand the concept of infinite limits. An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point.

Step 2: Analyze the given function, which is \( \frac{1}{3x} \). As \( x \) approaches 0 from the right (denoted by \( x \to 0^+ \)), the value of \( 3x \) becomes a very small positive number.

Step 3: Consider the behavior of the function \( \frac{1}{3x} \) as \( x \to 0^+ \). Since \( 3x \) is a small positive number, \( \frac{1}{3x} \) becomes a large positive number.

Step 4: Conclude that as \( x \to 0^+ \), \( \frac{1}{3x} \) tends towards positive infinity. Therefore, the limit is \( \infty \).

Step 5: Write the final expression for the limit: \( \lim_{x \to 0^+} \frac{1}{3x} = \infty \).

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the positive side.

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One-Sided Limits

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can happen when the function approaches a vertical asymptote. In the given limit, we need to determine whether the function approaches positive or negative infinity as x approaches 0 from the right.

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One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left (−) or the right (+). In this problem, we are specifically looking at the right-hand limit (x→0⁺), which means we only consider values of x that are greater than 0. This is crucial for determining the behavior of the function near the point of interest.

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One-Sided Limits

Pratica correlata

Domanda del libro di testo

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

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Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)

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Domanda del libro di testo

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

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Domanda del libro di testo

Using the Sandwich Theorem

a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?

Give reasons for your answer.

[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

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Domanda del libro di testo

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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Domanda del libro di testo

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.