# How do you find the asymptotes for #g(x) = (4x^2+11) /( x^2+8x−9)#?

H.A. @

V.A.s @

No S.A.

The rules for horizontal asymptotes:

The rule for vertical asymptotes:

The rule for slant asymptotes:

Because the degrees are the same, there is no slant asymptote.

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To find the asymptotes of the function ( g(x) = \frac{4x^2+11}{x^2+8x-9} ), we need to consider both vertical and horizontal asymptotes.

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator doesn't. For this function, the denominator ( x^2 + 8x - 9 ) factors into ( (x - 1)(x + 9) ), so vertical asymptotes occur at ( x = -9 ) and ( x = 1 ).

Horizontal Asymptote: To find the horizontal asymptote, we examine the behavior of the function as ( x ) approaches positive or negative infinity. Since both the numerator and the denominator are quadratic polynomials, we compare the degrees of the leading terms. In this case, the degrees are equal, so we divide the leading coefficients. The horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator, which is ( y = \frac{4}{1} = 4 ).

Therefore, the asymptotes for the function ( g(x) = \frac{4x^2+11}{x^2+8x-9} ) are:

- Vertical asymptotes: ( x = -9 ) and ( x = 1 )
- Horizontal asymptote: ( y = 4 )

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