Why is it impossible to have lim_(x->0) f(x) and lim_(f(x)->0)f(x) simultaneously exist for any of these graphs?

A) f(x) = 1/x^2
B) f(x) = -1/x^2
C) f(x) = 1/x
D) f(x) = -1/x

2 Answers
Feb 12, 2017

All four of these graphs have the x-axis as a horizontal asymptote as x-> pm infty and the y-axis as a vertical asymptote as x->0 from the right or left.

Feb 12, 2017

Well, by definition, a vertical asymptote is when at x -> a, y -> pmoo from either side and x never touches a. Similarly, a horizontal asymptote is when at x -> pmoo, y -> b from either side without ever reaching b.

For the function

y = c/x,

where c is a constant, if x->0, y -> pmoo from either side of x = 0, so you have a vertical asymptote. You can also find that as x -> pmoo, y -> 0 but doesn't get there.

But if you have x -> 0 and consequently y -> pmoo, you can't also have x -> pmoo so that y -> 0. It's not possible to approach both asymptotes at once because x cannot approach 0 and pmoo at the same time, and y cannot approach pmoo and 0 at the same time.

(Imagine trying to run to two different places at once; can't do it.)

Both kinds of asymptotes are on the graph, to be sure, but you can only approach one of those kinds of asymptotes at a time.