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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 2.45
2장, 문제 2.45

Determine the following limits.
lim w→∞ (ln w2) / (ln w3 + 1)

검증된 단계별 안내
1
Identify the limit expression: \( \lim_{w \to \infty} \frac{\ln w^2}{\ln w^3 + 1} \).
Simplify the expression: \( \ln w^2 = 2 \ln w \) and \( \ln w^3 = 3 \ln w \).
Rewrite the limit using the simplified expressions: \( \lim_{w \to \infty} \frac{2 \ln w}{3 \ln w + 1} \).
Factor out \( \ln w \) from the denominator: \( \lim_{w \to \infty} \frac{2 \ln w}{\ln w (3 + \frac{1}{\ln w})} \).
Cancel \( \ln w \) and evaluate the limit: \( \lim_{w \to \infty} \frac{2}{3 + \frac{1}{\ln w}} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
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주요 개념

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of a function as w approaches infinity. Understanding limits helps in analyzing the asymptotic behavior of functions and is crucial for evaluating expressions that may not be directly computable.
추천 영상:
05:50
One-Sided Limits

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in growth and decay problems. In the given limit, the natural logarithm of w raised to a power is involved, which simplifies to a multiplication of the exponent and ln(w), illustrating the properties of logarithms.
추천 영상:
05:18
Derivative of the Natural Logarithmic Function

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the context of the given limit, as it can simplify the evaluation of the logarithmic expressions.
추천 영상:
5:50
Power Rules
관련 실천
교과서 질문

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

f(−1)=−2f\(\left\)(-1\(\right\))=-2, f(1)=2f\(\left\)(1\(\right\))=2, f(0)=0f\(\left\)(0\(\right\))=0, limx→∞f(x)=1{\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)=1}}, limx→−∞f(x)=−1{\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)=-1}}

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교과서 질문

Evaluate each limit and justify your answer. 

lim x→5 ln 6(√x^2−16−3) / 5x−25

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교과서 질문

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=4x3+11−x3f\(\left\)(x\(\right\))=\(\frac{4x^3+1}{1-x^3}\)

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교과서 질문

Describe the end behavior of g(x) = e-2x.

408
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교과서 질문

Use a graph of f to estimate limx→af(x){\(\displaystyle\]\lim\)_{x\(\to\) a}f\(\left\)(x\(\right\))} or to show that the limit does not exist. Evaluate f(x) near x=ax=a to support your conjecture.

f(x)=x−2ln∣x−2∣f\(\left\)(x\(\right\))=\(\frac{x-2}{\ln\left|x-2\right|}\); a=2a=2

391
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교과서 질문

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x3+12x3+16x6+1f\(\left\)(x\(\right\))=\(\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}\)}

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