Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\ln \left(x^{2}+64\right) \). Find vertical asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is (are) \( x=\square \) (Use a comma to separate answers as needed.) B. There are no vertical asymptotes. Find horizontal asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote(s) is (are) \( y=\square \). (Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

To analyze the function \( f(x) = \ln(x^{2} + 64) \), we start by investigating the vertical asymptotes. The logarithm is undefined for non-positive arguments. Since \( x^2 + 64 \) is always greater than zero for all real \( x \) (as it reaches its minimum value of 64 when \( x = 0 \)), there are no vertical asymptotes. Now, for the horizontal asymptotes, we look at \( f(x) \) as \( x \) approaches positive or negative infinity. As \( |x| \to \infty \), \( x^2 \) dominates the expression, leading to \( f(x) \to \ln(x^2) \), which simplifies to \( 2\ln(|x|) \). Thus, \( f(x) \) goes to infinity and does not settle at a finite value, meaning there are no horizontal asymptotes either. So, to answer your queries: - B. There are no vertical asymptotes. - B. There are no horizontal asymptotes.