What are the values and types of the critical points, if any, of f(x) = (x-5)/(x-3)^2f(x)=x5(x3)2?

1 Answer
Nov 11, 2016

The vertical asymptote is x=3x=3
The horizontal asymptote is y=0y=0
The intercepts are (0,-5/9)(0,59) and (5,0)(5,0)

Explanation:

As we cannot divide by 00, the vertical asymptote is x=3x=3

The degree of the numerator is << the dergee of the denominator, there is no oblique asymptote.
lim_(x->-oo)f(x)=lim_(x->-oo)x/x^2=lim_(x->-oo)1/x=0^(-)
lim_(x->+oo)f(x)=lim_(x->+oo)x/x^2=lim_(x->+oo)1/x=0^(+)
The horizontal asymptote is y=0

The intercepts are
on the y-axis, x=0=>f(0)=-5/9
on the x-axis, y=0=>0=x-5=>x=5
graph{(x-5)/(x-3)^2 [-10, 10, -5.005, 4.995]}