Question

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Answer 1

Examine the limit of y as x approaches ±∞.
\[\lim_{x\to \infty}\dfrac{-4x^6+6x+3}{8x^6+9x+3} \\
\lim_{x\to \infty}\dfrac{-4+6x^{-5}+3x^{-6}}{8+9x^{-5}+3x^{-6}}\\
-\frac{4}{8}\\
-\frac{1}{2}\]
In this case, the limit as x approaches negative infinity is the same as when x approaches positive infinity:
\[\lim_{x\to -\infty}\dfrac{-4x^6+6x+3}{8x^6+9x+3}\\
\lim_{x\to -\infty}\dfrac{-4+6x^{-5}+3x^{-6}}{8+9x^{-5}+3x^{-6}}\\
-\frac{1}{2}\]
This tells you that y has one horizontal asymptote at y = -1/2.