How does this limit problem follow from limit rules?

[suffian]

[SOLVED] A limit problem

I just need some help showing how this limit systematically follows from the limit rules:

           # x     2
      1   #       t           1
lim  ---  #  ---------- dt = ---
x->0   3  #     2             3
      x  # 0   t  + 1

My first chain of thought led to breaking the expression up as follows:
1/x^2 * ( 1/x * Integral[0..x, t^2/(t^2+1)] )

Then I just kind of figured that the subexpression on the right was the average value of the function being integrated from 0..x and as x->0 the average value would approach x^2/(x^2+1), which led to:

1/x^2 * x^2/(x^2+1) = 1/(x^2+1)
which would approach one as x approached zero.

But clearly that's wrong (not surprisingly since I made a sketchy move in the middle) since the answer is one-third. Can anyone show me how to do this?

edit: possibly w/o actually integrating because this is an exercise in which you're expected to know the FTofC but not how to integrate that.

edit2: oh, not supposed to no l'hospital's rule either.

 

[Hurkyl]

Your first thought would have worked if you only had 1/x out front instead of 1/x³. (because your limit would essentially be a derivative)

So the trick is to rewrite it in a form in which the fundamental theorem of calculus applies! In particular, if you can do a substitution in the integral so the bounds of integration are from 0 to x³, then you can use your thought to evaluate the limit.

 

[suffian]

Okay, I think this works then.

I tried to change Integral[0..x, t²/(t²+1)] into Integral[0..x³, f(t)]

Integral[0..x, t²/(t²+1)] = Integral[0..x³, f(t)]
d/dx[ Integral[0..x, t²/(t²+1)] ] = d/dx[ Integral[0..x³, f(t)] ]
x²/(x² + 1) = f(x³).3x²
f(x³) = 1/(3(x²+1)), (x != 0)
f(x) = 1/(3(x^2/3+1))

so..
Limit[ x->0, 1/x³ Integral[0..x³, 1/(3(t^2/3+1))] ]

Which is the average value of the function inside the integral from 0..x³, which approaches f(0) as x approaches zero, which would be 1/3. I hope that's a sufficient way to solve the problem.

 

[Hurkyl]

That's an interesting way to change the limits! But you got the right answer in the end, I'm going to have to look at it and see why it works; I never thought to do it that way.

Incidentally, I was thinking doing the substitution t = s^1/3. t = x => s = x³, t = 0 => s = 0, and the integral became

∫_0..x³ s^2/3 / (s^2/3 + 1) * (1/3) s^-2/3 ds

which is precisely the integral you got.


Anyways, then you're left with the form:

L = lim_x->0 1/x³ &int_0..x³ g(s) ds

We are also permitted to substitute in the limit variable, and I will do so to make things simpler. x³ = y

L = lim_y->0 1/y &int_0..y g(s) ds

By the fundamental theorem of calculus, if G(s) is the antiderivative of g(s):

L = lim_y->0 (G(y) - G(0)) / y = G'(0) = g(0)

So that's how you rigorously justify your last step.

 

[suffian]

Thanks!

 

[Hurkyl]

I guess, for the sake of completeness, I should specify that when I changed the limit variable, I had to use a function that is continuous and invertible near x = 0 (I think that alone is sufficient to permit the operation). y(x) = x^1/3 satisfies that condition.