Question

How do you identify all asymptotes for `f(x)=(x^2-3x+2)/x`?

Answer 1

See below.

Explanation:

Vertical asymptotes occur where the function is undefined, for:

`(x^2-3x+2)/x`

This is undefined for `x=0` ( division by zero )

Vertical asymptote is the line: `x=0`

Notice that the degree of the numerator is greater than the degree of the denominator. In this case there is an oblique asymptote. This will be a line of the form `y=mx+b`. To find this line, we divide the numerator by the denominator. We only need to divide until we have the equation of a line.

`:.`

Dividing by `x`:

`(x^2/x-3x/x+2/x)/(x/x)=x-3+2/x`

So the oblique asymptote is the line:

`color(blue)(y=x-3)`

The graph confirms these findings: