Assessment: (5 points each item) 5. Show that the \( \lim _{x \rightarrow e} \ln x=1 \). 6. Show that the \( \lim _{x \rightarrow 0}\left(\frac{\sin t}{t} * \frac{1-\cos t}{t}\right) \)

To show that \( \lim _{x \rightarrow e} \ln x=1 \), we can substitute \( x = e \) directly into the natural logarithm function. Since \( \ln e = 1 \), it follows that as \( x \) approaches \( e \), the value of \( \ln x \) approaches 1. Thus, the limit holds true, confirming that \( \lim _{x \rightarrow e} \ln x = 1 \). For the limit \( \lim _{t \rightarrow 0}\left(\frac{\sin t}{t} * \frac{1-\cos t}{t}\right) \), we can independently evaluate the two components involved. As \( t \) approaches 0, \( \frac{\sin t}{t} \) approaches 1 and \( \frac{1-\cos t}{t} \) approaches 0 (with \( 1-\cos t \approx \frac{t^2}{2} \) for small values of \( t \)). Therefore, our limit collapses to \( 1 \cdot 0 = 0 \), confirming that \( \lim _{t \rightarrow 0}\left(\frac{\sin t}{t} * \frac{1-\cos t}{t}\right) = 0 \).