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Section 2.6 : Infinite Limits

For problems 1 – 6 evaluate the indicated limits, if they exist.

1. For \(\displaystyle f\left( x \right) = \frac{9}{{{{\left( {x - 3} \right)}^5}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to {3^{\, - }}} f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to {3^{\, + }}} f\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to 3} f\left( x \right)\)
   Solution
2. For \(\displaystyle h\left( t \right) = \frac{{2t}}{{6 + t}}\) evaluate,
   1. \(\mathop {\lim }\limits_{t \to \, - {6^{\, - }}} h\left( t \right)\)
   2. \(\mathop {\lim }\limits_{t \to \, - {6^{\, + }}} h\left( t \right)\)
   3. \(\mathop {\lim }\limits_{t \to \, - 6} h\left( t \right)\)
   Solution
3. For \(\displaystyle g\left( z \right) = \frac{{z + 3}}{{{{\left( {z + 1} \right)}^2}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{z \to \, - {1^{\, - }}} g\left( z \right)\)
   2. \(\mathop {\lim }\limits_{z \to \, - {1^{\, + }}} g\left( z \right)\)
   3. \(\mathop {\lim }\limits_{z \to \, - 1} g\left( z \right)\)
   Solution
4. For \(\displaystyle g\left( x \right) = \frac{{x + 7}}{{{x^2} - 4}}\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to {2^{\, - }}} g\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to {2^{\, + }}} g\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to 2} g\left( x \right)\)
   Solution
5. For \(\displaystyle h\left( x \right) = \ln \left( { - x} \right)\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to {0^{\, - }}} h\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to {0^{\, + }}} h\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to 0} h\left( x \right)\)
   Solution
6. For \(\displaystyle R\left( y \right) = \tan \left( y \right)\) evaluate,
   1. \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} R\left( y \right)\)
   2. \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} R\left( y \right)\)
   3. \(\mathop {\lim }\limits_{y \to {\textstyle{{3\pi } \over 2}}} R\left( y \right)\)
   Solution

For problems 7 & 8 find all the vertical asymptotes of the given function.

1. \(\displaystyle f\left( x \right) = \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}}\) Solution
2. \(\displaystyle g\left( x \right) = \frac{{ - 8}}{{\left( {x + 5} \right)\left( {x - 9} \right)}}\) Solution