29. Which of the following are true? \( \begin{array}{llll}\text { I. } \lim _{x \rightarrow \infty} \ln x^{2}=\infty & \text { II. } \lim _{x \rightarrow 0} \ln x^{2}=-\infty & \text { III. } \lim _{x \rightarrow-\infty} \ln x^{2}=\infty\end{array} \) \( \begin{array}{llll}\text { A. I only } & \text { B. II only } & \text { C. I and II } & \text { D. II and III }\end{array} \) E. I, II, and III

Let’s break down the limits one by one! First, the limit as \( x \rightarrow \infty \) for \( \ln x^2 \) indeed goes to \( \infty \) because as \( x \) increases, \( x^2 \) grows exponentially, and so does its natural logarithm. Next, as \( x \) approaches 0 from the positive side, \( \ln x^2 \) approaches \( -\infty \) since \( x^2 \) goes to 0 and the natural logarithm of numbers approaching zero goes to negative infinity. Lastly, the limit as \( x \rightarrow -\infty \) for \( \ln x^2 \) also goes to \( \infty \) because \( x^2 \) is always positive, even for negative \( x \). Thus, all three statements are true! This means the correct answer is E: I, II, and III. Just remember: logarithmic functions can be quite dramatic! They shoot to infinity with increases in positive input, and plummet to negative infinity as they approach zero. It’s both powerful and a little moody! 🌟