Math Is Fun Forum

yonski

Member

Registered: 2005-12-14

Posts: 67

Convergence

Hi there,
if I have f(x) = x√(x^2 + a^2) - x^2 , where a is a constant and is quite a bit smaller than x, how do I find out what it converges to?

If I put in values of x which are really massive compared to a then I get close to zero, which is obvious. But if I put in kind of mid-range values then it seems to settle around another value. For example, I have a=17 and I try x as 500, 1000, 1500, 2000, etc then f(x) stays really close to 144. It's confusing me lol.

Any help appreciated :-)

yonski

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Student: "What's a corollary?"
Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Convergence

That is a truly bizarre function. The attached image is a graph (made with Excel) of 10^x against f(x), where a=17.

The function starts off low and converges to 144.5, getting to within 0.00002 before it starts fluctuating.
When x ≈ 25,000, it ceases to be a strictly increasing function but it continues to converge anyway.
At x ≈ 100,000, it crosses the 144.5 and starts oscillating away from there, with the oscillations getting more wild as x increases.

Then at x ≈ 100,000,000, it does a weird kind of "piecewise increasing" pattern.
Finally, at x ≈ 1 thousand million, it suddenly drops to 0 and stays there.

I'm not sure how much of that is actually happening and how much is caused by Excel making rounding errors, but it's still a very strange graph nonetheless.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Convergence

Everything "weird" you see is being cause by the difference error.  Subtracting two terms which are almost equal.  Try graphing the first part (without the -x^2).  There are no anomalies.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Convergence

How do you know? The anomalies in this graph only start appearing when x>10000, which means x² > 10^8. Even the wildest anomalies (which only appear when x is in the millions) are only by about 100, so you wouldn't be able to observe whether or not they are there.

Also, it's clear that the function definitely stays near 144.5 for lowish values of x, but it's equally clear by elementary analysis that the limit of the function is 0. So there must be some kind of transition.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Convergence

Try graphing x√(x^2 + a^2) and x^2 on the same plot.  They stay apart from each other for a while, then gradually converge.  There is no weird behavior once you take the difference error out of the "equation".

But if you're going to use analysis techniques rather than "look at this graph", might as well go all out.  That function is clearly differentiable so long as x or a are nonzero.  Furthermore, this derivative has no solutions when it is set to 0.  Finally, it's second derivative is never 0 on the real line as well.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."