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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 2 - Limits문제 2.4.9c
2장, 문제 2.4.9c

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→−2 h(x)

검증된 단계별 안내
1
Step 1: Identify the type of limit problem. Since the problem involves a vertical asymptote at x = -2, we are dealing with a limit where the function approaches infinity or negative infinity.
Step 2: Understand the behavior of the function near the asymptote. As x approaches -2, the function h(x) will either increase or decrease without bound.
Step 3: Consider the direction of approach. Determine if you need to evaluate the limit from the left (x approaches -2 from the left, denoted as x → -2⁻) or from the right (x approaches -2 from the right, denoted as x → -2⁺).
Step 4: Analyze the graph of h(x) near x = -2. Observe whether the function values are increasing towards positive infinity or decreasing towards negative infinity as x approaches -2 from either side.
Step 5: Conclude the limit based on the behavior observed. If the function approaches positive infinity from both sides, the limit is positive infinity. If it approaches negative infinity, the limit is negative infinity. If the behavior differs from each side, the limit does not exist.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

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

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function h(x) has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, h(x) will either increase or decrease without bound.
추천 영상:
가이드 코스
3:40
Introduction to Cotangent Graph Example 1

Limits

A limit describes the behavior of a function as the input approaches a particular value. In the context of the question, evaluating the limit of h(x) as x approaches -2 involves determining what value h(x) approaches as x gets closer to -2, which is critical for understanding the function's behavior near its vertical asymptote.
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. For the limit lim x→−2 h(x), it is important to consider both the left-hand limit (as x approaches -2 from values less than -2) and the right-hand limit (as x approaches -2 from values greater than -2) to fully understand the behavior of h(x) near the asymptote.
추천 영상:
05:50
One-Sided Limits
관련 실천
교과서 질문

Complete the following steps for the given functions. 


c. Graph ff and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

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교과서 질문

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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교과서 질문

Determine whether the following statements are true and give an explanation or counterexample.


c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

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교과서 질문

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→^3− h(x)

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교과서 질문

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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교과서 질문

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

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