To find the limit of the given rational function as \( x \) approaches \(-\infty\), we should first identify the highest degree terms in the numerator and the denominator. The highest degree term in the numerator is \(4x^3\) and in the denominator is \(2x^2\). We can simplify the expression by dividing every term by \(x^2\), the highest power of \(x\) in the denominator, and then evaluate the limit as \(x\) approaches \(-\infty\).
To evaluate the limit \(\lim _{x \rightarrow-\infty} \frac{4 x^{3}-5 x}{2 x^{2}-2}\), we first identify the highest degree terms in the numerator and the denominator. The highest degree term in the numerator is \(4x^3\) and in the denominator is \(2x^2\).
Divide every term in the numerator and the denominator by \(x^2\), the highest power of \(x\) in the denominator:
\[
\frac{4x^3 - 5x}{2x^2 - 2} = \frac{\frac{4x^3}{x^2} - \frac{5x}{x^2}}{\frac{2x^2}{x^2} - \frac{2}{x^2}} = \frac{4x - \frac{5}{x}}{2 - \frac{2}{x^2}}
\]
As \(x \rightarrow -\infty\), the terms \(\frac{5}{x}\) and \(\frac{2}{x^2}\) approach 0. Therefore, the expression simplifies to:
\[
\lim_{x \rightarrow -\infty} \frac{4x - \frac{5}{x}}{2 - \frac{2}{x^2}} = \lim_{x \rightarrow -\infty} \frac{4x}{2} = \lim_{x \rightarrow -\infty} 2x
\]
Since \(x\) is approaching \(-\infty\), \(2x\) approaches \(-\infty\).
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