well, in general, f(∞) may not be meaningless. in the case of real (or even rational) numbers, we can set:

, IF this limit exists (well, technically, we should also have some other restrictions on f, but

normally continuity will suffice).

both the sine and cosine function oscillate. however, the "phase shift" between sine and cosine remains constant, no matter how large x becomes:

as functions they do not "approach each other". since calculators are necessarily finite and have a limit to how large a number they can store,

(which typically is actually an integer (my guess is 1 less than some power of 2), converted to scientific notation when too large to display),

my guess is that "default values" for f(x), when x exceeds the maximum value storeable, are pre-programmed in. most of these are probably not to be trusted.