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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 2.4.7b
2장, 문제 2.4.7b

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→1^+ f(x)

검증된 단계별 안내
1
Identify that the limit \( \lim_{x \to 1^+} f(x) \) involves approaching the point \( x = 1 \) from the right.
Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1 from the right, \( f(x) \) will tend towards either positive or negative infinity.
Examine the behavior of \( f(x) \) as \( x \to 1^+ \) by considering the values of \( f(x) \) for \( x \) slightly greater than 1.
Determine whether \( f(x) \) increases without bound (approaches \( +\infty \)) or decreases without bound (approaches \( -\infty \)) as \( x \to 1^+ \).
Conclude the analysis of the limit based on the direction in which \( f(x) \) tends as \( x \to 1^+ \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m

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 f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.
추천 영상:
가이드 코스
3:40
Introduction to Cotangent Graph Example 1

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specific value from one side only. The notation lim x→1^+ f(x) indicates the limit of f(x) as x approaches 1 from the right (values greater than 1). Understanding one-sided limits is crucial for analyzing functions with vertical asymptotes.
추천 영상:
05:50
One-Sided Limits

Limit Behavior Near Asymptotes

The limit behavior near vertical asymptotes is characterized by the function's tendency to increase or decrease without bound. For instance, if lim x→1^+ f(x) approaches positive infinity, it indicates that as x gets closer to 1 from the right, the function's values rise indefinitely. This behavior is essential for understanding the overall shape and characteristics of the graph.
추천 영상:
03:07
Cases Where Limits Do Not Exist
관련 실천
교과서 질문

The hyperbolic cosine function, denoted cosh(x)\(\cosh\)\(\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\)\(\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


b. Evaluate cosh(0)\(\cosh\)\(\left\)(0\(\right\)). Use symmetry and part (a) to sketch a plausible graph for y=cosh(x)y=\(\cosh\)\(\left\)(x\(\right\)).

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교과서 질문

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


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

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교과서 질문

Assume you invest \$250 at the end of each year for 10 years at an annual interest rate of rr. The amount of money in your account after 10 years is given by A(r)=250((1+r)10−1)rA\left(r\right)=\frac{250\left(\left(1+r\right)^{10}-1\right)}{r}. Assume your goal is to have \$3500 in your account after 10 years.


b. Use a calculator to estimate the interest rate required to reach your financial goal.

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교과서 질문

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

b. Determine the value of aa for which gg is continuous from the right at 11. 

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교과서 질문

Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .

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교과서 질문

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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