How do you find vertical, horizontal and oblique asymptotes for # (x^2 + 5x + 6)/(x+3)#?

Question

Anna Allen · Accepted Answer

#"no asymptotes"# #"factor the numerator"# #=((x+2)cancel((x+3)))/cancel((x+3))=x+2# #"the removal of the factor "(x+3)" indicates a removable"# #"discontinuity (hole) at "x=-3# #"The simplified version is a linear equation with "# #"no asymptotes"# graph{(x^2+5x+6)/(x+3) [-10, 10, -5, 5]}

Lincoln Anderson · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for the function $ \frac{x^2 + 5x + 6}{x + 3} $, we analyze the behavior of the function as $ x $ approaches certain values.

1. **Vertical Asymptote:** Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. Set the denominator $ x + 3 $ equal to zero and solve for $ x $ to find the vertical asymptote.

$$ x + 3 = 0 $$
   $$ x = -3 $$

Therefore, the vertical asymptote is $ x = -3 $.

2. **Horizontal Asymptote:** Horizontal asymptotes occur when the degree of the numerator and denominator are equal. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (y = 0). If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

In this case, the degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote.

3. **Oblique Asymptote:** An oblique (slant) asymptote occurs when the degree of the numerator is one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial division (long division or synthetic division) to divide the numerator by the denominator.

$ (x^2 + 5x + 6) \div (x + 3) $

Performing the division gives:

$$
   \begin{array}{r|l}
    x^2 + 5x + 6 & x + 3 \
    \hline
    x^2 + 3x \
    \hline
    2x + 6 \
   \end{array}
   $$

Therefore, the oblique asymptote is $ y = x + 2 $.

In summary, the vertical asymptote is $ x = -3 $, there is no horizontal asymptote, and the oblique asymptote is $ y = x + 2 $ for the function $ \frac{x^2 + 5x + 6}{x + 3} $.