Questions: Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. lim (x -> ∞) (6x^3 - 8x - 3)/(4x^2 - 4x - 5) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. lim (x -> ∞) (6x^3 - 8x - 3)/(4x^2 - 4x - 5) = B. The limit does not exist and is neither ∞ nor -∞.

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.

lim (x -> ∞) (6x^3 - 8x - 3)/(4x^2 - 4x - 5)

Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. lim (x -> ∞) (6x^3 - 8x - 3)/(4x^2 - 4x - 5) =
B. The limit does not exist and is neither ∞ nor -∞.

Transcript text: Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. \[ \lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x-3}{4 x^{2}-4 x-5} \] Select the correct choice below and, if necessary, fill in the answer box within your choice. A. $\lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x-3}{4 x^{2}-4 x-5}=$ B. The limit does not exist and is neither $\infty$ nor $-\infty$.

Solution

Solution Steps

Step 1: Analyze the Degrees of the Polynomials

Consider the limit \[ \lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x-3}{4 x^{2}-4 x-5}. \] The degree of the numerator $6x^3 - 8x - 3$ is 3, and the degree of the denominator $4x^2 - 4x - 5$ is 2.

Step 2: Compare the Degrees

Since the degree of the numerator (3) is greater than the degree of the denominator (2), we conclude that the limit will approach infinity as $x$ approaches infinity.

Step 3: State the Limit

Thus, we find that \[ \lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x-3}{4 x^{2}-4 x-5} = \infty. \]