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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.6.37
2장, 문제 2.6.37

Infinite Limits


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


lim x→0⁺ 1 / 3x

검증된 단계별 안내
1
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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
05:50
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.
추천 영상:
05:50
One-Sided Limits

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.
추천 영상:
05:50
One-Sided Limits
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교과서 질문

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) = x², x = -2

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Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 sin(1 − cos t) / (1 − cos t)

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교과서 질문

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (1 − cos 3x) / 2x

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교과서 질문

Theory and Examples


If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).

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교과서 질문

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(x² + 25) − √(x² − 1))

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교과서 질문

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ (2√x + x⁻¹) / (3x − 7)

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