Question

What are the asymptotes of the function?

f(x)=2x^2-4/x^2-7x-8
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Vertical asymptote:

heckpoint: Lessons 3-4

Horizontal asymptote

Answer 1

Vertical asymptote: x=0

Horizontal asymptote: no horizontal asymptote

Answer 2

The vertical asymptotes of f(x) are x=-4 and x=2. The function has no horizontal asymptote.

Step-by-step explanation:

Vertical asymptotes: These occur where the denominator of the function approaches zero but the numerator doesn't, causing the function to approach positive or negative infinity.

Factor the denominator: f(x) = 2x^2-4 / (x+4)(x-2).

Identify the non-canceling factors in the denominator: x+4 and x-2.

Therefore, the function has vertical asymptotes at x=-4 and x=2.

Horizontal asymptotes: These occur when the ratio of the leading coefficients of the numerator and denominator approaches a constant value as x approaches positive or negative infinity.

Analyze the degrees of the numerator (2) and denominator (2).

Since the degrees are equal, the function has no horizontal asymptote.

Therefore, the vertical asymptotes are x=-4 and x=2, and there is no horizontal asymptote.