1. **State the problem:** We have two functions $P(x)$ and $Q(x)$, both with degree 7, and their end behaviors are given: - $P(x) \to -\infty$ as $x \to +\infty$ and $P(x) \to +\infty$ as $x \to -\infty$. - $Q(x) \to -\infty$ as $x \to +\infty$ and $Q(x) \to +\infty$ as $x \to -\infty$. We want to find the end behavior of $R(x) = P(x) \cdot Q(x)$. 2. **Recall the rule for end behavior of polynomial products:** The end behavior of the product $R(x) = P(x)Q(x)$ depends on the leading terms of $P(x)$ and $Q(x)$. 3. **Analyze the leading terms:** Since both $P$ and $Q$ are degree 7 polynomials, their leading terms dominate for large $|x|$. Given $P(x) \to -\infty$ as $x \to +\infty$ and $P(x) \to +\infty$ as $x \to -\infty$, the leading coefficient of $P(x)$ is negative (because odd degree with negative leading coefficient behaves this way). Similarly, $Q(x)$ has the same end behavior, so its leading coefficient is also negative. 4. **Determine the leading term of $R(x)$:** The degree of $R(x)$ is $7 + 7 = 14$. The leading coefficient of $R(x)$ is the product of the leading coefficients of $P$ and $Q$. Since both leading coefficients are negative, their product is positive. 5. **Describe the end behavior of $R(x)$:** For an even degree polynomial with positive leading coefficient: - As $x \to +\infty$, $R(x) \to +\infty$. - As $x \to -\infty$, $R(x) \to +\infty$. **Final answer:** $$\boxed{\lim_{x \to \pm \infty} R(x) = +\infty}$$ This means $R(x)$ grows without bound positively at both ends of the $x$-axis.