convergent? help please.

[purpledragooon]

Suppose f is a continuous function on [0, infinite) and lim x-->infinite f(x)=1. is it possible that integral 0,infinite f(x)dx is convergent? Justify answer.

i believe it is but honestly no clue how to prove or even justify. teacher hinted to compare to g(x)=1/x

 

[pka]

Suppose f is a continuous function on [0, infinite) and lim x-->infinite F(x)=1. is it possible that integral 0,infinite f(x)dx is convergent? Justify answer.

Using the definition of limits:
\(\displaystyle 0.5>0\) so \(\displaystyle \exists N>0\) such that if \(\displaystyle x\ge N\) then \(\displaystyle |f(x)-1|<0.5\).

But that means \(\displaystyle 0.5<f(x)~[\forall x\ge N].\)

So \(\displaystyle \int_N^\infty {f(x)} > \int_N^\infty {0.5dx} \). SO?

 

[mmm4444bot]

Suppose f is a continuous function on [0, infinite) and lim x-->infinite F(x)=1. is it possible that integral 0,infinite f(x)dx is convergent? Justify answer.

Notes on terminology and symbolism:

When naming a function, do not use f and F as though they were interchangeable. Use one or the other because F is not the same function as f.

Also, the correct spelling is infinity.

Cheers :cool:

 

[purpledragooon]

Using the definition of limits:
\(\displaystyle 0.5>0\) so \(\displaystyle \exists N>0\) such that if \(\displaystyle x\ge N\) then \(\displaystyle |f(x)-1|<0.5\).

But that means \(\displaystyle 0.5<f(x)~[\forall x\ge N].\)

So \(\displaystyle \int_N^\infty {f(x)} > \int_N^\infty {0.5dx} \). SO?

to be honest i don't what half those symbols were, he gave a hint in a class to compare with g(x)=1/x.
i really need someone to show me how this is done

 

[mmm4444bot]

First of all, do you understand all of the given information (i.e., the behavior of f(x) and the meaning of 'convergent' with respect to the infinite integral of f over its domain)?

I think that this exercise is a thought experiment. You need to first understand the given scenario.

 

[purpledragooon]

First of all, do you understand all of the given information (i.e., the behavior of f(x) and the meaning of 'convergent' with respect to the infinite integral of f over its domain)?

I think that this exercise is a thought experiment. You need to first understand the given scenario.

to be honest i feel i understand but i don't have a grasp of what to do with it. he explained that a question like this was going to be on the test and i will definitly fail if i can't even figure out his example

 

[mmm4444bot]

i feel i understand

I'm not trying to be rude here, but I have no interest in your feelings.

Either you understand what it means for an integral to converge OR you do not.

We cannot know what you understand, until after you tell us.

PS: Do you know? Definite integrals give the area under the graph of f?

 

[pka]

to be honest i don't what half those symbols were, he gave a hint in a class to compare with g(x)=1/x. i really need someone to show me how this is done

Your instructor is being deceitful with you.
If I were you I would report his/her lack of understanding to the school. It seems that you have no idea about limits much less about the theory of integrals. If you don't it is a crime for you instructor to ask you this question. On the other hand, if you have simply not paid close attention to your notes then shame on you.

 

[HallsofIvy]

Your instructor is being deceitful with you.
If I were you I would report his/her lack of understanding to the school. It seems that you have no idea about limits much less about the theory of integrals. If you don't it is a crime for you instructor to ask you this question. On the other hand, if you have simply not paid close attention to your notes then shame on you.

Perhaps the point is that the instructor is not being deceitful and has taught all those things. As far as "compare with 1/x", since the given function has limit 1 as x goes to infinity, and this person's instructor probably expects him to know that 1/x goes to 0 and so, for some finite X, if x> X then f(x)> 1/x. Also that f(x)= 1/x is not integrable.

Or were you being sarcastic?

 

[HallsofIvy]

Also, the correct spelling is infinity.

Cheers :cool:

Specifically, "infinite" is an adjective, "infinity" is the noun.