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

Apr 15, 2018

See below.

#### Explanation:

Vertical asymptotes occur where the function is undefined, for:

$\frac{{x}^{2} - 3 x + 2}{x}$

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 $y = m x + b$. To find this line, we divide the numerator by the denominator. We only need to divide until we have the equation of a line.

$\therefore$

Dividing by $x$:

$\frac{{x}^{2} / x - 3 \frac{x}{x} + \frac{2}{x}}{\frac{x}{x}} = x - 3 + \frac{2}{x}$

So the oblique asymptote is the line:

$\textcolor{b l u e}{y = x - 3}$

The graph confirms these findings: