How do you find the Vertical, Horizontal, and Oblique Asymptote given # f(x) = (x^2) / (x-2)#?

1 Answer
Dec 29, 2016

Answer:

The vertical asymptote is #x=2#
The oblique asymptote is #y=x+2#
No horizontal asymptote

Explanation:

The domain of #f(x)# is #D_f(x)=RR-{2} #

As we cannot divide by #0#, #x!=2#

Therefore,

The vertical asymptote is #x=2#

As the degree of the numerator is #># than the degree of the denominator, we have an oblique asymptote.

Let's do a long division

#color(white)(aaaa)##x^2##color(white)(aaaaaaaa)##∣##x-2#

#color(white)(aaaa)##x^2-2x##color(white)(aaaa)##∣##x+2#

#color(white)(aaaaa)##0+2x#

#color(white)(aaaaaaa)##+2x-4#

#color(white)(aaaaaaaa)##+0+4#

So,
#f(x)=x+2+(4)/(x-2)#

Now we calculate the limits

#lim_(x->-oo)(f(x)-(x+2))=lim_(x->-oo)(4)/(x-2)=0^(-)#

#lim_(x->+oo)(f(x)-(x+2))=lim_(x->+oo)(4)/(x-2)=0^(+)#

So,

The oblique asymptote is #y=x+2#

graph{(y-(x^2)/(x-2))(y-x-2)=0 [-28.86, 28.86, -14.44, 14.44]}