Questions: Determine the following limits. Enter DNE if a limit fails to exist, except in case of an infinite limit. If an infinite limit exists, enter ∞ or -∞, as appropriate. - lim x→∞ (8x^2-4x^3-240+92x)/(-18x-3x^2-24)= - lim x→-∞ (8x^2-4x^3-240+92x)/(-18x-3x^2-24)=

Solution Steps

To determine the limits of the given rational functions as \( x \) approaches \( \infty \) and \( -\infty \), we need to analyze the degrees of the polynomials in the numerator and the denominator. The highest degree terms will dominate the behavior of the function as \( x \) approaches infinity or negative infinity.

1. For \( \lim_{x \rightarrow \infty} \frac{8 x^{2}-4 x^{3}-240+92 x}{-18 x-3 x^{2}-24} \):

   • Identify the highest degree terms in the numerator and the denominator.
   • Simplify the expression by dividing both the numerator and the denominator by the highest degree term in the denominator.
   • Evaluate the limit of the simplified expression as \( x \) approaches infinity.

2. For \( \lim_{x \rightarrow -\infty} \frac{8 x^{2}-4 x^{3}-240+92 x}{-18 x-3 x^{2}-24} \):

   • Follow the same steps as above, but consider the behavior as \( x \) approaches negative infinity.

To determine the limits, we first identify the highest degree terms in the numerator and the denominator of the function: \[ f(x) = \frac{-4x^3 + 8x^2 + 92x - 240}{-3x^2 - 18x - 24} \]

As \( x \) approaches \( \infty \) or \( -\infty \), the highest degree terms will dominate the behavior of the function. Therefore, we focus on the highest degree terms: \[ f(x) \approx \frac{-4x^3}{-3x^2} = \frac{-4}{-3}x = \frac{4}{3}x \]

For \( \lim_{x \to \infty} f(x) \): \[ \lim_{x \to \infty} \frac{4}{3}x = \infty \]

For \( \lim_{x \to -\infty} f(x) \): \[ \lim_{x \to -\infty} \frac{4}{3}x = -\infty \]

Final Answer