\( \begin{array}{lll}\text { 24. } \lim _{x \rightarrow \pi} \frac{\sin x}{x-\pi}= & \text { B. } 0 & \text { C. } \\ \begin{array}{lll}\text { A. }-1 & \text { D. } \pi & \text { E. The limit does not exis }\end{array}\end{array} \)

To evaluate the limit \( \lim_{x \rightarrow \pi} \frac{\sin x}{x - \pi} \), we can substitute \( x = \pi \) into the equation. However, this leads to \( \frac{0}{0} \), which is an indeterminate form. To resolve this, we can apply L'Hôpital's Rule, which involves taking the derivative of the numerator and denominator. The derivative of \(\sin x\) is \(\cos x\) and the derivative of \(x - \pi\) is \(1\). Thus, the limit becomes: \[ \lim_{x \rightarrow \pi} \frac{\cos x}{1} = \cos(\pi) = -1. \] So, the answer is A. \(-1\). For a little historical context, the concept of limits and indeterminate forms was rigorously developed in the 18th century by mathematicians like Augustin-Louis Cauchy. Before then, terms were rather vaguely used and often led to confusion in calculus. For real-world applications, understanding limits can help in fields like engineering and physics, especially when determining instantaneous rates of change or analyzing systems approaching certain conditions. For instance, during the design of a roller coaster, engineers may use limits to assess the maximum safe speed as the coaster approaches its peak, ensuring both safety and exhilaration!