To find the limit of the given function as \( x \) approaches infinity, we need to analyze the degrees of the polynomials in the numerator and the denominator. The highest degree term in the numerator is \( -3x^5 \) and in the denominator is \( x^2 \). As \( x \) approaches infinity, the term with the highest degree in the numerator will dominate, and similarly for the denominator. Therefore, the limit will be determined by the ratio of these leading terms.
To find the limit of the function as \( x \) approaches infinity, we first identify the leading terms in both the numerator and the denominator. The leading term in the numerator is \( -3x^5 \) and in the denominator is \( x^2 \).
As \( x \) approaches infinity, the highest degree terms will dominate the behavior of the function. Therefore, we can simplify the expression by focusing on these leading terms:
\[
\lim_{x \to \infty} \frac{-3x^5}{x^2}
\]
Simplify the fraction by dividing the terms:
\[
\frac{-3x^5}{x^2} = -3x^3
\]
Evaluate the limit of the simplified expression as \( x \) approaches infinity:
\[
\lim_{x \to \infty} -3x^3
\]
Since \( x^3 \) grows without bound as \( x \) approaches infinity, \( -3x^3 \) will approach negative infinity.
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