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Whenever I think about how a function behaves, I always try to identify a general pattern of behavior with some common numbers (somewhere between 5 and 100 maybe) and then I try to see if anything interesting happens around 1, 0 and into negative numbers if applicable.

If that all works out, I essentially assume that I know that the function is going to behave similarly for very large numbers as it does for those relatively small numbers.

Are there notable (famous, clever or common) functions where very large numbers would cause them to behave significantly differently than would initially be thought if I followed my regular experimental pattern? If so, are there any warning signs I should be aware of?

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You mean, you compute f(some x's ∈ [5, 100]), f(0), f(1), f(some x < 0) and try to deduce the behavior of f? –  KennyTM Aug 5 '10 at 20:23
That's a much more exacting way to think about it, but roughly yes. –  Nick Aug 5 '10 at 20:40
This is almost never a good idea. I am having a harder time coming up with examples where this does work, depending on what kind of "functions" you are looking at. For example, this is always a bad idea if the prime numbers are involved in any way. –  Qiaochu Yuan Aug 5 '10 at 21:56
Wow, some really great answers here. –  Nick Aug 6 '10 at 3:21
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4 Answers

up vote 6 down vote accepted

The Griewank function,

$$ f(\mathbf x) = \frac1{4000}\sum_{i=1}^n x_i^2 - \prod_{i=1}^n \cos\left(\frac{x_i}{\sqrt i}\right) + 1 $$

which is one of the objective functions used in testing optimization algorithms, looks completely different in large scale (dominated by x2) and small scale (dominated by cos x).

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Many rational functions $f(x)=\frac{n(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}$ (where deg(r) < deg(d)) behave very differently in the general vicinity of the zeros of d(x) than for large (positive or negative) values x. Near the zeros of d(x), the values of $\frac{r(x)}{d(x)}$ dominate the values of q(x) (that is, f(x) behaves like $\frac{r(x)}{d(x)}$), whereas for large (positive or negative) values of x, the values of q(x) dominate the values of $\frac{r(x)}{d(x)}$ (that is, f(x) behaves like q(x)).

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The Chebyshev bias behavior is properly understood only by examining up to very very large numbers.

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Polya's conjecture and its subsequent counterexample is of a similar nature. (I am quoting the relevant points from Wikipedia.) Based on experience from small numbers, Polya conjectured that among the first $n$ numbers, at least a half of them have an odd number of divisors. Haselgrove showed that this statement has a counterexample smaller than $2 \times 10^{361}$. In fact, the smallest counterexample is also quite big, $> 9 \times 10^8$. –  Srivatsan Aug 1 '11 at 19:29
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The values of a function $f(x)$ for relatively small values of its argument $x$ is typically a very bad predictor of asymptotic behavior of $f(x)$ for large $x$. This is true even when $f(x)$ is an analytic function which is uniquely determined by its values on any small interval $x\in[-\epsilon,\epsilon]$.

Have a look at this excerpt from "Concrete Mathematics" for a (not the worst possible) example of how deceptive the "small argument values" intuition could be.

It helps to cultivate an expansive attitude when we're doing asymptotic analysis: We should think big, when imagining a variable that approaches infinity. For example, the hierarchy says that $\log n\prec n^{0.0001}$; this might seem wrong if we limit our horizons to teeny-tiny numbers like one googol, $n = 10^{100}$. For in that case, $\log n = 100$, while $n^{0.0001}$ is only $10^{0.01}\approx 1.0233$. But if we go up to a googolplex, $n = 10^{10^{100}}$, then $\log n = 10^{100}$ pales in comparison with $n^{0.0001} = 10^{10^{96}}$.

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