Odd Degree Limit B5E8A0

1. **Problem Statement:** We are given a polynomial function $f(x)$ of odd degree such that $$\lim_{x \to \infty} f(x) = -\infty.$$ We need to find $$\lim_{x \to -\infty} f(x).$$ 2. **Recall the behavior of odd degree polynomials:** - A polynomial of odd degree has the general form: $$f(x) = a_n x^n + \dots + a_1 x + a_0,$$ where $n$ is odd and $a_n \neq 0$. - The end behavior of $f(x)$ is dominated by the leading term $a_n x^n$. - For odd $n$, as $x \to \infty$, $a_n x^n$ tends to $\infty$ if $a_n > 0$ and to $-\infty$ if $a_n < 0$. - As $x \to -\infty$, $a_n x^n$ tends to $-\infty$ if $a_n > 0$ and to $\infty$ if $a_n < 0$. 3. **Given:** $$\lim_{x \to \infty} f(x) = -\infty,$$ which means the leading coefficient $a_n$ is negative. 4. **Therefore, as $x \to -\infty$:** Since $n$ is odd and $a_n < 0$, the leading term behaves as $$a_n (-\infty)^n = a_n \cdot (-\infty) = \infty,$$ because an odd power preserves the sign of $x$, and multiplying by negative $a_n$ flips the sign. 5. **Conclusion:** $$\lim_{x \to -\infty} f(x) = \infty.$$