How do you find vertical, horizontal and oblique asymptotes for #(x^2 - 2x + 3) / x#?

Question

Serenity Jones · Accepted Answer

Vertical asymptotes:

Vertical asymptotes occur when the denominator of a rational function equals to 0 (this being because division by 0 is undefined in mathematics). We can find any vertical asymptotes by setting the denominator to 0 and solving.

#x = 0#

There will be a vertical asymptote at #x = 0#

Horizontal asymptotes:

Horizontal asymptotes only occur when the degree of the denominator is higher or equal to that of the numerator. We don't have this situation in our function.

Oblique asymptotes:

Oblique asymptotes occur when the denominator has a lower degree than the numerator. If the function is #f(x) = (g(x))/(h(x))#, there will be an oblique asymptote at the quotient of #g(x)/ (h(x))#.

Therefore, we will have to divide your rational function. A thorough understanding of division of polynomials is usually a pre-requisite to finding oblique asymptotes.

By synthetic division:

#"0_| 1 -2 3"#
#" 0 0 0"#
#"---------------"#
#" 1 -2 3"#

The quotient is therefore #x - 2#, with the remainder being #3#.

There will therefore be an oblique asymptote at #y = x - 2#

Here is the graph of the function:

Hopefully this helps!

Cora Alexander · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for the function $ \frac{x^2 - 2x + 3}{x} $, follow these steps:

1. Vertical asymptotes:
   Set the denominator equal to zero and solve for $ x $: $ x = 0 $. This is the equation of the vertical asymptote.

2. Horizontal asymptote:
   If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. In this case, the degree of the numerator is 2, and the degree of the denominator is 1, so there is no horizontal asymptote.

3. Oblique asymptote (if exists):
   Divide the numerator by the denominator using polynomial long division or synthetic division. The quotient obtained represents the equation of the oblique asymptote, if it exists.

Let's perform polynomial long division to find the oblique asymptote:
$$ \frac{x^2 - 2x + 3}{x} = x - 2 + \frac{3}{x} $$
The quotient obtained is $ x - 2 $, which represents the equation of the oblique asymptote.

Therefore, for the function $ \frac{x^2 - 2x + 3}{x} $:
- Vertical asymptote: $ x = 0 $
- Horizontal asymptote: None
- Oblique asymptote: $ y = x - 2 $