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Describe the long run behavior of \( f(x)=(x-4)^{2}(x+3)^{3}(x-3)^{2} \) As \( x \rightarrow-\infty, f(x) \rightarrow \) ? As \( x \rightarrow \infty, f(x) \rightarrow \) ?

Ask by Vaughan Ramirez. in the United States
Mar 17,2025

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As \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity. As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity.

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As \( x \rightarrow -\infty \), the dominant term in the polynomial \( f(x) = (x-4)^2(x+3)^3(x-3)^2 \) is \( (x+3)^3 \), since it raises \( x \) to the highest power. Therefore, \( f(x) \) will approach \( -\infty \) because the power of the leading term is odd and negative. So, \( f(x) \rightarrow -\infty \). As \( x \rightarrow \infty \), all factors of the polynomial grow larger as \( x \) increases. Since they are all raised to even powers or combined, the function approaches \( +\infty \). Thus, \( f(x) \rightarrow +\infty \).

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