Q:

Function behavior with very large variables

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?

A:

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).

(source: geatbx.com)

(source: geatbx.com)

(source: geatbx.com)

