Question

determine the following limit. (limlimits_{x \to -infty} (5x^3 - 6x^2 + 1)) select the correct choice and, if necessary, fill in the answer box to complete your choice. (\bigcirc) a. (limlimits_{x \to -infty} (5x^3 - 6x^2 + 1) = square) (simplify your answer.) (\bigcirc) b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

Step1: Analyze the leading term

For the polynomial \(5x^3 - 6x^2 + 1\), the leading term is \(5x^3\) (the term with the highest degree).

Step2: Evaluate the limit of the leading term as \(x\to -\infty\)

As \(x\to -\infty\), we consider the behavior of \(x^3\). When \(x\) is a large negative number, \(x^3\) is negative (since the cube of a negative number is negative). And multiplying by \(5\) (a positive coefficient), we have \(\lim_{x\to -\infty} 5x^3=-\infty\) (because the leading term dominates the behavior of the polynomial as \(x\) approaches \(\pm\infty\)). The other terms (\(- 6x^2\) and \(1\)) become negligible compared to \(5x^3\) as \(x\to -\infty\). So \(\lim_{x\to -\infty}(5x^3 - 6x^2 + 1)=\lim_{x\to -\infty}5x^3=-\infty\)

Answer:

A. \(\lim\limits_{x\to -\infty} (5x^3 - 6x^2 + 1)=\boxed{-\infty}\)