Describe the end behavior. In \( x \) on \( (0, \infty) \) A. \( \lim _{x \rightarrow 0^{+}} \ln x=1 \) and \( \lim _{x \rightarrow \infty} \ln x=\infty \) B. \( \lim _{x \rightarrow 0^{+}} \ln x=-\infty \) and \( \lim _{x \rightarrow \infty} \ln x=\infty \) \( \lim _{x \rightarrow 0^{+}} \ln x=-\infty \) and \( \lim _{x \rightarrow \infty} \ln x=0 \) \( \lim _{x \rightarrow 0^{+}} \ln x=0 \) and \( \lim _{x \rightarrow \infty} \ln x=1 \) D. \( x=1 \)

To analyze the end behavior of the natural logarithm function \( \ln x \) on the interval \( (0, \infty) \), we need to evaluate the limits as \( x \) approaches \( 0^{+} \) and \( \infty \). 1. **As \( x \rightarrow 0^{+} \)**: - The natural logarithm \( \ln x \) approaches negative infinity. This is because the logarithm of a number that is very close to zero (but positive) becomes very large in the negative direction. - Therefore, \( \lim_{x \rightarrow 0^{+}} \ln x = -\infty \). 2. **As \( x \rightarrow \infty \)**: - The natural logarithm \( \ln x \) increases without bound as \( x \) increases. This means that as \( x \) gets larger and larger, \( \ln x \) also gets larger and larger. - Therefore, \( \lim_{x \rightarrow \infty} \ln x = \infty \). Based on this analysis, the correct end behavior of \( \ln x \) on the interval \( (0, \infty) \) is: **B.** \( \lim_{x \rightarrow 0^{+}} \ln x = -\infty \) and \( \lim_{x \rightarrow \infty} \ln x = \infty \).