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

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 $f \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)}$, there will be an oblique asymptote at the quotient of $g \frac{x}{h \left(x\right)}$.

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.

$\text{0_| 1 -2 3}$
$\text{ 0 0 0}$
$\text{---------------}$
$\text{ 1 -2 3}$

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!