Question

How do you find all the asymptotes for `(3x^2) / (x^2 - 9)`?

Answer 1

Horizontal asymptote is `y=3`

Vertical asymptotes are `x=+-3`

Explanation:

Write as `y=(3x^2)/(x^2(1-9/x^2)) = 3/(1-9/x^2)`

`color(blue)("Determine horizontal asymptote")`

As `x` becomes increasingly large then `9/x^2` becomes increasingly small.

`color(blue)(y=lim_(x->+-oo) 3/(1-9/x^2) ->3/1 larr "horizontal asymptote")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Determine vertical asymptote")`

`color(brown)("As "x^2" gets closer and closer to 9 then "1-9/x^2" becomes ")` `color(brown)("smaller and smaller. So "3/(1-9/x^2)" becomes larger and larger.")`

`" "y=lim_(x->3) 3/(1-9/x^2)->oo`

`color(brown)("However; if "x=color(white)()^+3" then "9/(color(white)()^+3)<1`

`" "y=lim_(x->color(white)()^+3) 3/(1-9/x^2)->+oo`

`color(brown)("However; if "x=color(white)()^-3" then "9/(color(white)()^+3)>1`

`" "y=lim_(x->color(white)()^+3) 3/(1-9/x^2)->-oo`

So `color(blue)(y=lim_(x->3) 3/(1-9/x^2)->+-oo larr " vertical asymptotes")`