Question

What are the asymptotes of `y=1/(x-2)+1` and how do you graph the function?

Answer 1

Vertical: `x=2`
Horizontal: `y=1`

Explanation:

1. Find the vertical asymptote by setting the value of the denominator(s) to zero.
   `x-2=0` and therefore `x=2`.
2. Find the horizontal asymptote, by studying the end behavior of the function. The easiest way to do so is to use limits.
3. Since the function is a composition of `f(x)=x-2` (increasing) and `g(x)=1/x+1` (decreasing), it is decreasing for all defined values of `x`, i.e. `(-oo,2]uu[2,oo)`. graph{1/(x-2)+1 [-10, 10, -5, 5]}

`lim_(x->oo)1/(x-2)+1=0+1=1`
Other examples:
What is the zeros, degree and end behavior of `y=-2x(x-1)(x+5)`?