Taylor series vs. asymptotes for function approximations
A conversation with Ei. about the rational functions review he will do in his remedial calculus class today (and the rest of this week, though he doesn't know that yet) led to the following observation:
You can make an asymptote that looks like just about any function. Add your favorite function that doesn't converge to 0, A(x), to 1/x, and as x gets large the 1/x part becomes negligible, so you get a function that approaches A(x). To me this looks like a good way to approximate A(x) around any sufficiently large x_0 - not on a computer of course, because the computer would first have to approximate A(x) by a Taylor series before it could render A(x) + 1/x, defeating the purpose - but for analytic attacks on differential equations. You could add whatever function that converges to 0 best suits your purposes, to cancel things or get A(x) in a certain desired form. Is this something I should have learned in Elementary ODEs, or probably will learn in Advanced ODEs?
^_^
~If you can't join 'em, beat 'em.
You can make an asymptote that looks like just about any function. Add your favorite function that doesn't converge to 0, A(x), to 1/x, and as x gets large the 1/x part becomes negligible, so you get a function that approaches A(x). To me this looks like a good way to approximate A(x) around any sufficiently large x_0 - not on a computer of course, because the computer would first have to approximate A(x) by a Taylor series before it could render A(x) + 1/x, defeating the purpose - but for analytic attacks on differential equations. You could add whatever function that converges to 0 best suits your purposes, to cancel things or get A(x) in a certain desired form. Is this something I should have learned in Elementary ODEs, or probably will learn in Advanced ODEs?
^_^
~If you can't join 'em, beat 'em.