Let
𝑓
,
𝑔
be positive, continuous functions with domain
𝑅
.

Let
𝐿
=
lim
𝑥
→
∞
𝑓
(
𝑥
)
𝑔
(
𝑥
)
.

Assume
∫
1
∞
𝑓
(
𝑥
)
𝑑
𝑥
is convergent.

Which of the following statements must be true?

Select all the correct answers.

Question

Let 
𝑓
,
𝑔
  be positive, continuous functions with domain 
𝑅
  .

Let 
𝐿
=
lim
𝑥
→
∞
𝑓
(
𝑥
)
𝑔
(
𝑥
)
 .

Assume 
∫
1
∞
𝑓
(
𝑥
)
𝑑
𝑥
 is convergent.

Which of the following statements must be true?

Select all the correct answers.

BlackTom AI · Accepted Answer

We start by parsing the given setup: f and g are positive, continuous on R, and L = lim_{x→∞} f(x)/g(x). It is also given that ∫_1^∞ f(x) dx converges. We will evaluate each statement in light of these facts and general limit comparison ideas.

Option 1: IF L DNE, THEN ∫_1^∞ g(x) dx is not convergent.
This claim tries to link the nonexistence of the limit of f/g to the divergence of the integral of g. However, the absence of a limit does not force divergence of ∫ g. It is entirely possible for f and g to behave in ways that prevent a limit from existing while still having ∫ g converge or diverge independently. For example, one could have oscillatory behavior in f/g that prevents a limit, yet g's tail could be integrable. Therefore, this statement is not guaranteed to be true in all scenarios; it is not a must.

Option 2: IF L exists, THEN ∫_1^∞ g(x) dx is convergent.
Here we consider the case when the limit L exists and is finite. If L > 0, then f ~ L g for large x, so convergence of ∫ f implies convergence of ∫ g (since multiplying by a positive constant preserves convergence of improper integrals). If L = 0, however, f is little-o of g, meaning f is much smaller than g asymptotically. In that case, the convergence of ∫ f does not force the convergence of ∫ g; g could be large enough on a tail to cause divergence even though f remains small enough to be integrable. Because the statement would have to hold even when L = 0, it is not guaranteed to be true. Thus this option is not must.

Option 3: IF L = 3, THEN ∫_1^∞ g(x) dx is convergent.
This is a straightforward application of the limit comparison principle. If L = 3 (a positive finite limit), then for large x we have f(x) ≈ 3 g(x). Since ∫ f(x) dx converges, the comparison f ≈ 3 g implies ∫ g must also converge; multiplying a function by a positive constant does not affect convergence. Therefore, this statement must be true.

Option 4: IF L is ∞, THEN ∫_1^∞ g(x) dx is convergent.
If L = ∞, then f(x)/g(x) → ∞, so for any large M there exists X such that for x > X we have f(x) ≥ M g(x). In particular, for large x, f(x) dominates g(x) by an unbounded factor. Since ∫_1^∞ f(x) dx converges, the tail integral ∫_X^∞ f(x) dx is finite. But for x > X we have f(x) ≥ M g(x) with arbitrarily large M, forcing ∫_X^∞ g(x) dx to be bounded above by (1/M) ∫_X^∞ f(x) dx, which tends to zero as M grows. This implies ∫_1^∞ g(x) dx must converge as well. Hence this statement is also must be true.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 56 (12.9,12.10)

Let 𝑓 , 𝑔 be positive, continuous functions with domain 𝑅 . Let 𝐿 = lim 𝑥 → ∞ 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) . Assume ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent. Which of the following statements must be true? Select all the correct answers.

查看解析

登录即可查看完整答案

类似问题

Let 𝑓 be a positive, continuous function with domain 𝑅 . Assume ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is divergent. Which of the following improper integrals must also be divergent? Select all the correct answers.

Let 𝑓 be a positive, continuous function with domain 𝑅 . Assume ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent. Which of the following improper integrals must also be convergent? Select all the correct answers.

Consider an improper integral ∫ 1 ∞ 1 𝑥 2 𝑎 − 1 𝑑 𝑥 . Which condition should 𝑎 satisfy to make this improper integral convergent? Select the best answer (which includes as many values as possible).

Which of the following integrals are IMPROPER? Select all the correct answers.

Which of the following statements is correct? Note: only one of them is correct.

In Question 0, select all of the integrals that DIVERGE.

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单