\( \operatorname{Lim}_{n \rightarrow \infty}\left(\frac{5 n+1}{5 n}\right)^{n}=: \) \( \begin{array}{llll}\text { A) } e^{\frac{1}{5}} & \text { (B) } e^{5} & \text { (C) } e^{-5} & \text { (D) } e^{\frac{2}{5}}\end{array} \)

To solve the limit \( \operatorname{Lim}_{n \rightarrow \infty}\left(\frac{5 n+1}{5 n}\right)^{n} \), we simplify the expression inside the limit: \[ \frac{5 n + 1}{5 n} = 1 + \frac{1}{5n} \] Now, we rewrite the limit: \[ \operatorname{Lim}_{n \rightarrow \infty} \left(1 + \frac{1}{5n}\right)^{n} \] Using the fact that \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \), we can set \( x = \frac{1}{5} \). Thus, we have: \[ \operatorname{Lim}_{n \rightarrow \infty} \left(1 + \frac{1}{5n}\right)^{n} = e^{\frac{1}{5}} \] Therefore, the answer is: **A) \( e^{\frac{1}{5}} \)** Now, a little context: With the roots of calculus and series, the expression we've dealt with here often gives insight into exponential growth patterns. Many limits like this show up in understanding processes that can be approximated by continuous growth, such as compound interest, population growth, or certain physical phenomena like radioactive decay. Additionally, if you're curious about deeper explorations, the concept of limits often leads to studying derivatives and integrals, where you get to play with the rate of change and area under curves. Books like "Calculus" by James Stewart or "Introduction to Real Analysis" by Bartle and Sherbert provide fantastic insights. Happy reading!