Estimate the Limit by Graphing the Function for an Appropriate

question 86

Essay

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule.
Show each step of your calculation.
-A student attempted to use l'Hôpital's Rule as follows. Identify the student's error.
$\begin{aligned}\lim _{ x \rightarrow \infty } \frac { \sin ( 1 / x ) } { e ^ { 1 / x } } = & \lim _{ x \rightarrow \infty } \frac { - x ^ { - 2 } \cos ( 1 / x ) } { - x ^ { - 2 } e ^ { 1 / x } } \\& = \lim _ { x \rightarrow \infty } \frac { \cos ( 1 / x ) } { e ^ { 1 / x } } = \frac { 1 } { 1 } = 1\end{aligned}$