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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 2.6.42
2장, 문제 2.6.42

Infinite Limits


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


lim x→−5⁻ (3x) / (2x + 10)

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Identify the type of limit: This is a one-sided limit as x approaches -5 from the left, denoted as x → -5⁻.
Analyze the denominator: As x approaches -5 from the left, the expression 2x + 10 approaches 0. Specifically, since x is slightly less than -5, 2x + 10 will be slightly negative.
Analyze the numerator: The numerator 3x approaches 3(-5) = -15 as x approaches -5.
Determine the sign of the fraction: Since the numerator approaches -15 (a negative value) and the denominator approaches a small negative value, the overall fraction (3x)/(2x + 10) approaches a positive value.
Conclude the limit: As x approaches -5 from the left, the fraction (3x)/(2x + 10) approaches positive infinity, denoted as ∞.

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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. Limits can be finite or infinite, and they are essential for defining derivatives and integrals.
추천 영상:
05:50
One-Sided Limits

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as x→a⁻) or the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may have different behaviors on either side of a point, particularly at points of discontinuity.
추천 영상:
05:50
One-Sided Limits

Infinite Limits

An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point. This is indicated by writing the limit as ∞ or −∞. Understanding infinite limits is important for analyzing vertical asymptotes and the overall behavior of functions near points where they may not be defined.
추천 영상:
05:50
One-Sided Limits
관련 실천
교과서 질문

At what points are the functions in Exercises 13–30 continuous?

y = (2x – 1)¹/³

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교과서 질문

Theory and Examples


Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

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교과서 질문

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

289
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교과서 질문

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find


b. limx→c 2f(x)g(x)

241
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교과서 질문

Using the Sandwich Theorem


a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?


Give reasons for your answer.


[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

380
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교과서 질문

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


g(x) = 1/(2 + (1/x))

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