Describe the vertical asymptote(s) and hole(s) for the graph of \( y=\frac{(x-3)(x+8)}{(x+8)(x+2)} \) (1 point) asymptote: \( x=-8 \) and hole: \( x=-2 \) asymptote: \( x=2 \) and hole: \( x=8 \) asymptote: \( x=-2 \) and hole: none asymptote: \( x=-2 \) and hole: \( x=-8 \)

To find the vertical asymptote(s) and hole(s) for the function \( y=\frac{(x-3)(x+8)}{(x+8)(x+2)} \), we start by factoring the numerator and the denominator. The factor \( (x+8) \) cancels out, which indicates a hole at \( x = -8 \). The remaining denominator \( (x+2) \) does not cancel and leads to a vertical asymptote at \( x = -2 \). So, we have: Asymptote: \( x = -2 \) and hole: \( x = -8 \). Exploring the historical development of functions and asymptotes, mathematicians like Augustin-Louis Cauchy in the 19th century popularized the study of limits and the behavior of functions approaching certain values. They laid the groundwork for understanding asymptotic behavior, which is crucial in fields like calculus. For real-world applications, asymptotes can help in modeling various phenomena, such as population growth and supply-demand curves. They give insight into the behavior of a function as it approaches certain limits, which is vital for predicting trends and making informed decisions in economics and engineering.