How do you find the vertical, horizontal or slant asymptotes for f(x)= (x+3)/(x-3)?

1 Answer
Jun 19, 2016

Vertical asymptote is x=3
Horizontal asymptote is y=1

Explanation:

Tony B

The equation becomes undefined at x=3 in the denominator as 3-3=0. Basically this means that mathematically you are not allowed to divide by 0.

color(blue)("Vertical asymptotes")

lim_(xto3^(+))(x+3)/(x-3) -> (x+3)/(0^+)=+oo

lim_(Xto3^-)(x+3)/(x-3)-> (x+3)/(0^-)=-oo

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color(blue)("Horizontal asymptotes")

As x becomes bigger and bigger then the addition or subtraction of 3 becomes insignificant. Consequently we end up with basically x/x

lim_(xtooo^+) (x+3)/(x-3) ->(+oo)/(+oo) = +1

lim_(xtooo^-) (x+3)/(x-3)->(-oo)/(-oo) = +1

So the horizontal asymptote is +1
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Check: using polynomial division.

(x+3)-:(x-3) = 1+6/(x-3)

lim_(x->3^(+-) ) = 1+-oo

lim_(x->oo^(+-) ) = 1+-0=1