QuestionAugust 24, 2025

Evaluate the following limit. lim _(xarrow infty )(-4x^3-3x^2-4x+8)/(-3x^2)-x+2 square

Evaluate the following limit. lim _(xarrow infty )(-4x^3-3x^2-4x+8)/(-3x^2)-x+2 square
Evaluate the following limit. lim _(xarrow infty )(-4x^3-3x^2-4x+8)/(-3x^2)-x+2 square

Solution
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Answer

The limit is undefined (approaches negative infinity). Explanation 1. Identify the highest power of ( ( \(x\) ) ) The highest power in the numerator is \(x^3\) and in the denominator is \(x^2\). 2. Divide each term by \(x^3\) Rewrite the expression as \(\frac{-4 - \frac{3}{x} - \frac{4}{x^2} + \frac{8}{x^3}}{-\frac{3}{x} - \frac{1}{x^2} + \frac{2}{x^3}}\). 3. Evaluate the limit as \(x \to \infty\) As \(x \to \infty\), terms with ( ( \(x\) ) ) in the denominator approach zero. Thus, the expression simplifies to \(\frac{-4}{0}\). 4. Conclude the behavior of the limit Since the denominator approaches zero while the numerator remains constant, the limit does not exist in a finite sense.

Explanation

1. Identify the highest power of ( \(x\) )<br /> The highest power in the numerator is \(x^3\) and in the denominator is \(x^2\).<br /><br />2. Divide each term by \(x^3\)<br /> Rewrite the expression as \(\frac{-4 - \frac{3}{x} - \frac{4}{x^2} + \frac{8}{x^3}}{-\frac{3}{x} - \frac{1}{x^2} + \frac{2}{x^3}}\).<br /><br />3. Evaluate the limit as \(x \to \infty\)<br /> As \(x \to \infty\), terms with ( \(x\) ) in the denominator approach zero. Thus, the expression simplifies to \(\frac{-4}{0}\).<br /><br />4. Conclude the behavior of the limit<br /> Since the denominator approaches zero while the numerator remains constant, the limit does not exist in a finite sense.
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