Section 2.8 : Limits at Infinity, Part II
11. Evaluate \(\displaystyle \mathop {\lim }\limits_{t \to \,\infty } {\tan ^{ - 1}}\left( {\frac{{4 + 7t}}{{2 - t}}} \right)\).
Show SolutionFirst notice that,
\[\mathop {\lim }\limits_{t \to \,\infty } \frac{{4 + 7t}}{{2 - t}} = - 7\]If you aren’t sure about this limit you should go back to the previous section and work some of the examples there to make sure that you can do these kinds of limits.
Then answer is then,
\[\mathop {\lim }\limits_{t \to \,\infty } {\tan ^{ - 1}}\left( {\frac{{4 + 7t}}{{2 - t}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\tan }^{ - 1}}\left( { - 7} \right)}}\]Do not get so used the “special case” limits that we tend to usually do in the problems at the end of a section that you decide that you must have done something wrong when you run across a problem that doesn’t fall in the “special case” category.