another curly one tossed at me by my partner

Is it possible to conceive of the following in mathematical terms:

a periodic/cyclic function tending to an ever increasingly straight line, reaching infinitely close, before becoming equivalent to y=x.

in other words, is it possble to arrive at a straight line by other means than y=x, such that, given specified proportions, numbers, magnitudes, one could create a formula to produce a straight line without recourse to y=x?

y=2x | y=0 | x=0 | 5x+3y=7 are all straight lines
y=sin(x)-(sin(x)-x) is the same as y=x.
y=1/2((x+1)^2 - x^2 - 1) is the same as y=x

I have no idea what you are looking for, maybe that helps.