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

1 Answer
Jun 8, 2016

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#

#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:

enter image source here

Hopefully this helps!