How do you find the asymptotes for #g(x)=(3x^2+2x-1)/(x^2-4)#?

Question

Elijah Allen · Accepted Answer

#x=+-2 and y = 0#,

Expressed in partial fractions, #y = g(x) = 3 +(5/2)/(x-2)+(3/2)/(x+2)# This form readily discloses that #x=+-2# represent the vertical asymptotes. Also, as #x to +-oo, y to 0#. And so, y = 0 represents the vertical asymptote. Illustrative graph is inserted.I graph{y(x^2-4)-3x^2-2x+1=0 [-40, 40, -20, 20]}

Theodore Amidon · Answer

undefined To find the asymptotes of the function $g(x) = \frac{3x^2 + 2x - 1}{x^2 - 4}$, we first need to identify the vertical asymptotes, horizontal asymptotes, and slant (or oblique) asymptotes.

1. Vertical Asymptotes: Vertical asymptotes occur where the denominator of the rational function is equal to zero and the numerator is not zero. In this case, the denominator $x^2 - 4$ is zero when $x = 2$ or $x = -2$. Therefore, the vertical asymptotes are $x = 2$ and $x = -2$.

2. Horizontal Asymptotes: Horizontal asymptotes can be found by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line $y = 0$. If the degrees are equal, the horizontal asymptote is the ratio of 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 2. So, to find the horizontal asymptote, we look at the ratio of the leading coefficients: $3/1 = 3$. Therefore, the horizontal asymptote is $y = 3$.

3. Slant (Oblique) Asymptotes: Slant or oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator. To find the slant asymptote, we perform polynomial long division or use other methods to divide the numerator by the denominator.

$$ \frac{3x^2 + 2x - 1}{x^2 - 4} = 3 + \frac{14x - 1}{x^2 - 4} $$

As $x$ approaches infinity or negative infinity, the term $ \frac{14x - 1}{x^2 - 4} $ approaches zero, so the slant asymptote is the line $y = 3$.

In summary, the asymptotes for $g(x) = \frac{3x^2 + 2x - 1}{x^2 - 4}$ are:
- Vertical asymptotes at $x = 2$ and $x = -2$.
- Horizontal asymptote at $y = 3$.
- Slant asymptote at $y = 3$.