How do you find the asymptotes for # (2x-10)/(x^2+3x+2)#?

Question

Elias Ackerman · Accepted Answer

vertical asymptotes x = -1 , x =- 2
horizontal asymptote y = 0

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s let the denominator equal zero. solve: # x^2 + 3x + 2 = 0 → (x + 1 )(x + 2) = 0 # #rArr x = -1" and " x = -2" are the asymptotes "# Horizontal asymptotes occur as #lim_(xto+-oo) f(x) to 0 # When the degree of the numerator < degree of denominator , as is the case here then the equation is always y = 0 Here is the graph of the function. graph{(2x-10)/(x^2+3x+2) [-10, 10, -5, 5]}

Liam Brennan · Answer

undefined To find the asymptotes of the rational function $ \frac{{2x - 10}}{{x^2 + 3x + 2}} $, you need to determine the vertical asymptotes, horizontal asymptotes, and oblique asymptotes (if any).

1. Vertical asymptotes:
   Vertical asymptotes occur where the denominator of the rational function is equal to zero and the numerator is not zero. Set the denominator $ x^2 + 3x + 2 $ equal to zero and solve for $ x $. The solutions represent the vertical asymptotes.

$ x^2 + 3x + 2 = 0 $  
   $ (x + 1)(x + 2) = 0 $  
   $ x = -1 $ or $ x = -2 $

So, the vertical asymptotes are $ x = -1 $ and $ x = -2 $.

2. Horizontal asymptotes:
   Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. If the degrees are the same, divide the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In this case, the degree of the numerator is 1 and the degree of the denominator is 2. So, there is a horizontal asymptote.

Divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal asymptote.

The horizontal asymptote is $ y = 0. $

3. Oblique asymptotes:
   Oblique asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial long division or use other methods to divide the numerator by the denominator. The quotient represents the oblique asymptote.

Since the degree of the numerator is less than the degree of the denominator, there are no oblique asymptotes.

In summary:
- Vertical asymptotes: $ x = -1 $ and $ x = -2 $
- Horizontal asymptote: $ y = 0 $