Find the horizontal asymptote of the graph of y = start fraction negative 4 x superscript 6 baseline plus 6 x plus 3 over 8 x superscript 6 baseline plus 9 x plus 3 end fraction.
A. y = 1
B. y = –one-half
C. y = 0
D. There is no horizontal asymptote.
To find the horizontal asymptote, we need to look at the highest degree terms in the numerator and the denominator. In this case, both the numerator and the denominator have a term of x^6. So, as x approaches infinity or negative infinity, these terms will dominate and the function will behave like:
y = (-4x^6)/(8x^6) = -1/2
Therefore, the horizontal asymptote is y = -1/2, option B.
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y = (-4x^6 + 6x + 3)/(8x^6 + 9x + 3)
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y = (-4x^6 + 6x + 3)/(8x^6 + 9x + 3)
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To find the horizontal asymptote of the given function, we need to analyze the behavior of the function as x approaches positive or negative infinity.
First, let's simplify the function by dividing both the numerator and denominator by the highest power of x, which is x^6:
y = (-4x^6 + 6x + 3) / (8x^6 + 9x + 3)
Now, as x approaches infinity, the x^6 terms dominate the other terms in both the numerator and denominator. So, we can ignore the other terms except for the x^6 terms:
y ≈ (-4x^6) / (8x^6)
When we simplify this, we get:
y ≈ -1/2
Therefore, as x approaches positive or negative infinity, the function approaches the horizontal asymptote y = -1/2.
So, the answer is B. y = -one-half.