Solving Integration Question Using L'Hopital's Rule - Step by Step Guide

[peripatein]

Hi,

Homework Statement

I'd appreciate some help in computing the integral in the attachment.

Homework Equations

The Attempt at a Solution

I presume it should be handled using L'hopital, but I am not sure how to demonstrate that that is indeed the case.

 

[CAF123]

Yes, I'Hopital will be applied. Consider using the fundamental theorem of Calculus as well.

 

[peripatein]

But L'hopital applies for 0/0, for instance. How is that the case here?

 

[CAF123]

But L'hopital applies for 0/0, for instance. How is that the case here?

As x → 0, the upper limit tends to 0 and since the lower limit is already zero, we know $\int_{a}^{a} f dt = 0$. ( and the function f in your example is continuous near x = 0)

 

[peripatein]

Is it rigorous/formal enough, to simply state that int [a,a] f dt = 0?

 

[CAF123]

Is it rigorous/formal enough, to simply state that int [a,a] f dt = 0?

I would say: $$\int_0^0 f\,dt = F(0) - F(0) = 0,$$ since a function cannot have one to many as an option.

If I recall correctly, and perhaps someone could confirm, it is not possible to evaluate that integral (analytically) if the limits are finite (i.e something like 0 to $a$, $a$ a real number).

 

[peripatein]

Is it true then that the limit diverges?

 

[CAF123]

Is it true then that the limit diverges?

I wouldn't figure so quickly. So we have justified using i'Hopital. Now apply i'Hopital.
(the bit about your integrand not being integrable with finite limits was just for interest. It doesn't really relate to this problem since the region of int. is [0,0] and provided an integral exists this is always zero if those are your bounds).

 

[SammyS]

Is it true then that the limit diverges?

I don't think so, but I hurried through it, so I could have made a mistake.

 

[peripatein]

No, it is equal to zero. Thanks!

 

[SammyS]

No, it is equal to zero. Thanks!

That's what I got too ... zero.