Determine the end behavior of f(x) = (-4x^3 + 7x + 9)/(8x^6 - 9x^4 - 2x) as x tends to minus infinity and as x tends to infinity.

Step-by-step solution

1. 1

   Compare degrees before doing anything else. The numerator has degree $3$ and the denominator degree $6$. Whenever the denominator wins by any margin, the quotient is squeezed toward $0$ at both ends — the lower-order terms cannot change that.

2. 2

   Keep only the leading terms. For large $|x|$, $f(x)\approx\frac{-4x^3}{8x^6}=\frac{-1}{2x^3}.$ Formally, divide numerator and denominator by $x^6$; every surviving term except the constant $8$ in the denominator goes to $0$.

3. 3

   Take $x\to+\infty$. Then $x^3\to+\infty$, so $\dfrac{-1}{2x^3}\to 0$ through negative values: $f(x)\to 0^{-}$.

4. 4

   Take $x\to-\infty$. Then $x^3\to-\infty$, so $\dfrac{-1}{2x^3}\to 0$ through positive values: $f(x)\to 0^{+}$.

5. 5

   State the conclusion. Both limits are $0$, so $y=0$ is a horizontal asymptote on both sides. Numerically $f(10^{6})\approx-5\times10^{-19}$ and $f(-10^{6})\approx+5\times10^{-19}$, matching the sign analysis.