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.

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To analyze the function $$ f(x) = \ln(x^2 + 64) $$, we will follow these steps:

1. **Identify the domain**: The logarithmic function is defined for positive arguments. Therefore, we need to find when $$ x^2 + 64 > 0 $$.

- Since $$ x^2 $$ is always non-negative and $$ 64 $$ is positive, $$ x^2 + 64 $$ is always positive for all real $$ x $$. Thus, the domain of $$ f(x) $$ is all real numbers, $$ (-\infty, \infty) $$.

2. **Find vertical asymptotes**: Vertical asymptotes occur where the function approaches infinity or negative infinity as $$ x $$ approaches a certain value. Since $$ f(x) $$ is defined for all $$ x $$ in its domain and does not approach infinity at any finite $$ x $$, there are no vertical asymptotes.

3. **Find horizontal asymptotes**: Horizontal asymptotes are found by evaluating the limits of $$ f(x) $$ as $$ x $$ approaches infinity or negative infinity.

- As $$ x \to \infty $$:
     $$
     f(x) = \ln(x^2 + 64) \approx \ln(x^2) = 2\ln(x) \to \infty
     $$
   - As $$ x \to -\infty $$:
     $$
     f(x) = \ln(x^2 + 64) \approx \ln(x^2) = 2\ln(-x) \to \infty
     $$

Since $$ f(x) $$ approaches infinity in both directions, there are no horizontal asymptotes.

4. **Sketch the graph**: The graph of $$ f(x) = \ln(x^2 + 64) $$ will be a continuous curve that increases without bound as $$ x $$ moves away from zero in both the positive and negative directions.

Now, summarizing the findings:

- **Vertical asymptote(s)**: 
  - B. There are no vertical asymptotes.

- **Horizontal asymptote(s)**: 
  - B. There are no horizontal asymptotes.

Thus, the final answers are:
- Vertical asymptote(s): B. There are no vertical asymptotes.
- Horizontal asymptote(s): B. There are no horizontal asymptotes.
- **Vertical asymptote(s)**: B. There are no vertical asymptotes. - **Horizontal asymptote(s)**: B. There are no horizontal asymptotes.

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