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Section 2.6 : Infinite Limits

For problems 1 – 8 evaluate the indicated limits, if they exist.

1. For \(\displaystyle g\left( x \right) = \frac{{ - 4}}{{{{\left( {x - 1} \right)}^2}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to {1^{\, - }}} g\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to {1^{\, + }}} g\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to 1} g\left( x \right)\)
2. For \(\displaystyle h\left( z \right) = \frac{{17}}{{{{\left( {4 - z} \right)}^3}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{z \to \,{4^{\, - }}} h\left( z \right)\)
   2. \(\mathop {\lim }\limits_{z \to \,{4^{\, + }}} h\left( z \right)\)
   3. \(\mathop {\lim }\limits_{z \to \,4} h\left( z \right)\)
3. For \(\displaystyle g\left( t \right) = \frac{{4{t^2}}}{{{{\left( {t + 3} \right)}^7}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{t \to \, - {3^{\, - }}} g\left( t \right)\)
   2. \(\mathop {\lim }\limits_{t \to \, - {3^{\, + }}} g\left( t \right)\)
   3. \(\mathop {\lim }\limits_{t \to \, - 3} g\left( t \right)\)
4. For \(\displaystyle f\left( x \right) = \frac{{1 + x}}{{{x^3} + 8}}\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to \, - {2^{\, - }}} f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \, - {2^{\, + }}} f\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to \, - 2} f\left( x \right)\)
5. For \(\displaystyle f\left( x \right) = \frac{{x - 1}}{{{{\left( {{x^2} - 9} \right)}^4}}}\) evaluate,
   1. \(\mathop {\lim }\limits_{x \to \,{3^{\, - }}} f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \,{3^{\, + }}} f\left( x \right)\)
   3. \(\mathop {\lim }\limits_{x \to \,3} f\left( x \right)\)
6. For \(W\left( t \right) = \ln \left( {t + 8} \right)\) evaluate,
   1. \(\mathop {\lim }\limits_{t \to \, - {8^{\, - }}} W\left( t \right)\)
   2. \(\mathop {\lim }\limits_{t \to \, - {8^{\, + }}} W\left( t \right)\)
   3. \(\mathop {\lim }\limits_{t \to \, - 8} W\left( t \right)\)
7. For \(h\left( z \right) = \ln \left| z \right|\) evaluate,
   1. \(\mathop {\lim }\limits_{z \to {0^{\, - }}} h\left( z \right)\)
   2. \(\mathop {\lim }\limits_{z \to {0^{\, + }}} h\left( z \right)\)
   3. \(\mathop {\lim }\limits_{z \to 0} h\left( z \right)\)
8. For \(R\left( y \right) = \cot \left( y \right)\) evaluate,
   1. \(\mathop {\lim }\limits_{y \to \,{\pi ^{\, - }}} R\left( y \right)\)
   2. \(\mathop {\lim }\limits_{y \to \,{\pi ^{\, + }}} R\left( y \right)\)
   3. \(\mathop {\lim }\limits_{y \to \,\pi } R\left( y \right)\)

For problems 9 – 12 find all the vertical asymptotes of the given function.

1. \(\displaystyle h\left( x \right) = \frac{{ - 6}}{{9 - x}}\)
2. \(\displaystyle f\left( x \right) = \frac{{x + 8}}{{{x^2}{{\left( {5 - 2x} \right)}^3}}}\)
3. \(\displaystyle g\left( t \right) = \frac{{5t}}{{t\left( {t + 7} \right)\left( {t - 12} \right)}}\)
4. \(\displaystyle g\left( z \right) = \frac{{{z^2} + 1}}{{{{\left( {{z^2} - 1} \right)}^5}{{\left( {z + 15} \right)}^6}}}\)