Question

Will mark brainliest and give 20 points! In what ways can vertical, horizontal, and oblique asymptotes be identified? Please use your own example to identify.

Answer 1

Vertical asymptote:
When you have a rational expression in which the denominator is zero, you have a vertical asymptote. So to find vertical asymptotes, just set the denominator of your rational expression equal to zero, and then, solve for $x$:
$\frac{x-1}{x-3}$
Set the denominator equal to zero:
$x-3=0$
Solve for $x$:
$x=3$ is the vertical asymptote of our rational expression.

Horizontal asymptote:
Here we have two scenarios.
1) Is the degree of the denominator is higher than the degree of the numerator, you will have a horizontal asymptote at $y=0$:
$y= \frac{x-1}{x^{2}+3}$
Since the degree of the denominator is higher of the degree of the numerator, our rational expression will have an asymptote at $y=0$
2) If the degree of both denominator and numerator is the same, the rational expression will have an horizontal asymptote at the ratio of the leading coefficients:
$\frac{3x^{2}+5}{2x^2-3x+1}$
Leading coefficients: 3 and 2
Ratio of leading coefficients:
$\frac{3}{2}$. Our rational expression will have an horizontal asymptote at $y= \frac{3}{2}$

Oblique asymptote:
If the degree of the numerator is higher than the degree of the numerator, you will have an oblique asymptote. To find it, we are going to perform long division; the quotient (without the remainder) will be the equation of the oblique asymptote line:
$\frac{x^2+5x+2}{x+1}$
The quotient of the long division is $x-1$ with a remainder of 2; therefore, the equation of the oblique asymptote line will be:
$y=x+4$