Explain how to sind the value Sor each limit b) \( \lim _{x \rightarrow-\infty} \frac{4 x^{4}-8 x^{6}+9}{6 x+5 x^{5}+2} \)

To find the limit as \( x \) approaches negative infinity in the expression \( \frac{4 x^{4}-8 x^{6}+9}{6 x+5 x^{5}+2} \), you should first identify the term with the highest degree in both the numerator and the denominator. In the numerator, the dominating term is \( -8 x^{6} \), and in the denominator, it is \( 5 x^{5} \). Next, divide all terms in the numerator and the denominator by \( x^{6} \), yielding \( \frac{4/x^{2} - 8 + 9/x^{6}}{6/x^{5} + 5 + 2/x^{6}} \). As \( x \) approaches negative infinity, the terms with \( 1/x^{n} \) where \( n > 0 \) will approach zero. Thus, the limit simplifies to \( \frac{-8}{5} \), which equals \( -\frac{8}{5} \) as \( x \) goes to negative infinity. This tells us that the polynomial growth in the negative direction directly influences the limit. For a solid understanding of limits, practice with a variety of rational functions where you can apply similar principles. Exploring common types, like polynomials or exponential functions, helps build confidence and identifies the dominant terms faster. Analyzing limits can become intuitive with time, so play around with different types of functions to see the behavior as \( x \) approaches positive and negative infinity!