Question

determine the following limit. (limlimits_{x \to infty} (2x^3 - 8x^2 + 1)) select the correct choice below and, if necessary, fill in the answer box to complete your choice. (\bigcirc) a. (limlimits_{x \to infty} (2x^3 - 8x^2 + 1) = square) (\bigcirc) b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

Step1: Analyze the leading term

For the polynomial \(2x^3 - 8x^2 + 1\), the leading term (the term with the highest power of \(x\)) is \(2x^3\). When finding the limit as \(x\to\infty\), the behavior of the polynomial is determined by its leading term.

Step2: Evaluate the limit of the leading term

As \(x\to\infty\), the term \(2x^3\) will also approach \(\infty\) because the cube of a large positive number is a large positive number, and multiplying by 2 (a positive constant) keeps it positive and growing without bound. The other terms (\(-8x^2\) and \(1\)) become negligible compared to \(2x^3\) as \(x\) becomes very large. So, \(\lim_{x\to\infty}(2x^3 - 8x^2 + 1)=\lim_{x\to\infty}2x^3=\infty\).

Answer:

A. \(\lim\limits_{x\to\infty}(2x^3 - 8x^2 + 1)=\infty\)