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Section 2.8 : Limits at Infinity, Part II

12. Evaluate \(\displaystyle \mathop {\lim }\limits_{w \to \,\infty } {\tan ^{ - 1}}\left( {\frac{{3{w^2} - 9{w^4}}}{{4w - {w^3}}}} \right)\).

Show Solution

First notice that,

\[\mathop {\lim }\limits_{w \to \,\infty } \frac{{3{w^2} - 9{w^4}}}{{4w - {w^3}}} = \infty \]

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.

Now, recalling Example 7 from this section, we know that because the argument goes to infinity in the limit the answer is,

\[\mathop {\lim }\limits_{w \to \,\infty } {\tan ^{ - 1}}\left( {\frac{{3{w^2} - 9{w^4}}}{{4w - {w^3}}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{\pi }{2}}}\]