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Your friend was asked to evaluate \[ \lim_{x\rightarrow \infty} \frac{x^2 + \sin{x}}{x^2}. \] They thought it seems like a clear case of L'H\^{o}pital's rule, and so here is the work:
\[ \begin{split}
    & \lim_{x\rightarrow \infty} \frac{x^2 + \sin{x}}{x^2} \qquad \left(\frac{\infty}{\infty}\right) \\
    =& \lim_{x \rightarrow \infty} \frac{2x + \cos{x}}{2x} \qquad \left(\frac{\infty}{\infty}\right) \\
    =& \lim_{x\rightarrow \infty} \frac{2 - \sin{x}}{2}
\end{split} \]
and since $\lim_{x\rightarrow \infty} \frac{2 - \sin{x}}{2}$ does not exist, neither does the original limit. However, when you plot $\frac{x^2 + \sin{x}}{x^2}$, it sure looks like the limit exists.

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    \includegraphics[width=0.6\textwidth]{mcr3.png}
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\begin{enumerate}
    \item Find the error in your friend's reasoning.
    \item Provide a correct solution. What is the actual limit?
\end{enumerate}


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