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Section 2.6 : Infinite Limits
6. For \(\displaystyle R\left( y \right) = \tan \left( y \right)\) evaluate,
- \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} R\left( y \right)\)
- \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} R\left( y \right)\)
- \(\mathop {\lim }\limits_{y \to {\textstyle{{3\pi } \over 2}}} R\left( y \right)\)
Hint : Don’t forget the graph of the tangent function.
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a Show SolutionThe easiest way to do this problem is from the graph of the tangent function so here is a quick sketch of the tangent function over several periods.
From the sketch we can see that,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} \tan \left( y \right) = \infty }}\]b Show Solution
From the graph in the first part we can see that,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} \tan \left( y \right) = - \infty }}\]c Show Solution
From the first two parts that,
\[\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} R\left( y \right) \ne \mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} R\left( y \right)\]and so, from our basic limit properties we can see that \(\mathop {\lim }\limits_{y \to {\textstyle{{3\pi } \over 2}}} R\left( y \right)\) does not exist.