Asymptotes of (1-x^2)/(x^3)?

Question

I'm trying to find the asymptotes of m(x) = (1-x^2)/(x^3)

x = 0 is a vertical asymptote.

& horizontal asymptote is y= 0

But im confused because when i draw the graph using calculator the function crosses the line y=0 which is supposed to be an asymptote!

Why is that?

Answer 1

the calc is drawing the graph as a single, continuous line. I think it always does this (mine does it) and you just have to ignore what you know does not exist.

Answer 2

a horizontal asymtope is just a guide to show how a function behaves as x gets to infinity. A function can actually cross its horizontal asymptote.

Answer 3

sometimes the standard techniques of evaluating asymptotes don't really work for some rational functions

also these supposed "asymptotes" only reflect the end behaviour of the functions near negative and positive infinity not near zero

that is why at y= 0 the function actually crosses

x= 1 and x = -1

but after that then the asymptotes is accurately depicted since the graph will not touch the y=0 line again

always remember that these asymptotes reflect the behaviour as x approaches infinity and negative infinity since the standard technique to finding horizontal asymptotes is to calculates the limit of x approaching infinity and negative infinity

that is why you see most discrepancies near x=0 since the standard procedure is not to evaluate asymptotes using limit approaching zero

in fact, it is actually OK to cross an asymptote line near the origin, they are only asymptote when x is very close to infinity or negative infinity. One key thing to remember is that asymptote lines are not asymptotic in the middle, near the origin!  

hope this helps