How do you find the vertical, horizontal and slant asymptotes of: #(3x-2) / (x+1)#?

Question

Elias Ackerman · Accepted Answer

vertical asymptote x = -1
horizontal asymptote y = 3

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero. solve : x + 1 = 0 → x = -1 , is the asymptote Horizontal asymptotes occur as # lim_(x to +- oo) , f(x) to 0 # divide terms on numerator/denominator by x # ((3x)/x - 2/x)/(x/x + 1/x) = (3 - 2/x)/(1 + 1/x) # as #x to +- oo , y to (3-0)/(1+0) # #rArr y = 3 " is the asymptote " # Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes. graph{(3x-2)/(x+1) [-10, 10, -5, 5]}

Christian Antunez · Answer

undefined To find the vertical asymptote, set the denominator equal to zero and solve for $ x $.

For $ x + 1 = 0 $:
$$ x = -1 $$

This gives the vertical asymptote at $ x = -1 $.

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, divide the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

In this case, the degrees of the numerator and denominator are both 1. So, the horizontal asymptote is determined by dividing the leading coefficients.

Leading coefficient of the numerator: 3
Leading coefficient of the denominator: 1

So, the horizontal asymptote is $ y = \frac{3}{1} = 3 $.

To find the slant asymptote (oblique asymptote), if the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division. The quotient obtained will be the equation of the slant asymptote.

In this case, the degree of the numerator is one more than the degree of the denominator, so we'll perform polynomial long division:

$$ \frac{3x - 2}{x + 1} = 3 - \frac{5}{x + 1} $$

The equation of the slant asymptote is $ y = 3 $.