How does this limit problem follow from limit rules?

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In summary, the conversation discusses how to solve a limit problem using the limit rules and the fundamental theorem of calculus. The participants go through different approaches and eventually arrive at the solution of 1/3. They also discuss the justification for using a substitution in the integral and changing the limit variable.
  • #1
suffian
[SOLVED] A limit problem

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

Code:
           # 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.
 
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  • #2
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.
 
  • #3
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(x2/3+1))

so..
Limit[ x->0, 1/x3 Integral[0..x3, 1/(3(t2/3+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.
 
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  • #4
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 = s1/3. t = x => s = x3, t = 0 => s = 0, and the integral became

∫0..x3 s2/3 / (s2/3 + 1) * (1/3) s-2/3 ds

which is precisely the integral you got.


Anyways, then you're left with the form:

L = limx->0 1/x3 &int0..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 = limy->0 1/y &int0..y g(s) ds

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

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

So that's how you rigorously justify your last step.
 
  • #5
Thanks!
 
  • #6
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) = x1/3 satisfies that condition.
 

1. What is a limit problem?

A limit problem is a mathematical concept used to describe the behavior of a function as its input approaches a specific value. It is used to determine the value that a function gets closer to as its input gets closer to a certain value.

2. Why are limits important?

Limits are important because they help us understand the behavior of a function and its values. They are used in various fields, such as calculus, physics, and statistics, to solve real-world problems and make predictions.

3. How do you solve a limit problem?

To solve a limit problem, you need to follow a set of rules and techniques. These include factoring, simplifying, using algebraic manipulations, and applying special limits formulas. It is also important to understand the concept of continuity and the properties of limits.

4. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit refers to the behavior of a function as its input approaches a certain value from only one direction, either from the left or the right. A two-sided limit, on the other hand, considers the behavior of a function as its input approaches a certain value from both the left and the right.

5. Can a limit problem have more than one solution?

No, a limit problem can only have one solution. The limit represents the value that a function approaches, but it may never actually reach that value. It is possible for different functions to have the same limit, but the functions themselves are not the same.

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