5. Explain how to find the value for each limin a) \( \lim _{x \rightarrow-\infty}-\frac{2 x^{4}+8 x^{5}+7}{3 x^{5}+3 x^{3}-1} \)

To find the limit as \( x \) approaches negative infinity for the given expression, focus on the leading terms of both the numerator and denominator, since they dominate the behavior of the function. Here, the highest degree term in the numerator is \( 8x^5 \) and in the denominator, it's \( 3x^5 \). As \( x \to -\infty \), the limit simplifies to \( \lim_{x \to -\infty} -\frac{8x^5}{3x^5} \), which simplifies further to \( \lim_{x \to -\infty} -\frac{8}{3} \), resulting in \( -\frac{8}{3} \). Remember to always check for opportunities to simplify before diving deep into calculations. In this case, recognizing that the highest powers in both numerator and denominator influence the limit allows us to arrive at the answer quickly. Additionally, keeping track of signs is essential; something as simple as forgetting that \( x \) is negative could lead to confusion in determining the final answer.