# How do you find the vertical, horizontal or slant asymptotes for #f(x) = (x^3-8)/(x^2-5x+6)#?

##### 1 Answer

#### Explanation:

The function

**Vertical asymptotes**:

In order to work out whether a rational function, **vertical asymptotes**, we simply set the **denominator equal to #0#**. If we can

**solve**the equation, then we have vertical asymptotes, if not, then we don't.

In this case:

Using any method to solve this equation tells us that

**Horizontal asymptotes**

**Horizontal asymptotes** occur when the polynomial of the denominator of a rational function **has a higher degree** than the polynomial of the numerator. If so, then the ** #x#-axis** will be the horizontal asymptote. (The horizontal asymptote may change via translation)

In this function, this doesn't occur, so there are no horizontal asymptotes.

**Slant asymptotes**

**Slant asymptotes** occur when the polynomial of the denominator of a rational function **has a lower degree** than the polynomial of the numerator. In order to find our slant asymptote, we must **divide the numerator by the denominator**.

If we divide the numerator by the denominator, we get the slant asymptote as

And here's your graph plotted on Desmos. (Although it seems like the graph crosses the first horizontal asymptote, the graph is actually undefined for that part.)