Section 2.2 : The Limit

1. For the function \(\displaystyle f\left( x \right) = \frac{{8 - {x^3}}}{{{x^2} - 4}}\) answer each of the following questions.
   1. Evaluate the function at the following values of \(x\) compute (accurate to at least 8 decimal places).
      1. 2.5
      2. 2.1
      3. 2.01
      4. 2.001
      5. 2.0001
      1. 1.5
      2. 1.9
      3. 1.99
      4. 1.999
      5. 1.9999
   2. Use the information from (a) to estimate the value of \(\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{8 - {x^3}}}{{{x^2} - 4}}\).
   Solution
2. For the function \(\displaystyle R\left( t \right) = \frac{{2 - \sqrt {{t^2} + 3} }}{{t + 1}}\) answer each of the following questions.
   1. Evaluate the function at the following values of \(t\) compute (accurate to at least 8 decimal places).
      1. -0.5
      2. -0.9
      3. -0.99
      4. -0.999
      5. -0.9999
      1. -1.5
      2. -1.1
      3. -1.01
      4. -1.001
      5. -1.0001
   2. Use the information from (a) to estimate the value of \(\displaystyle \mathop {\lim }\limits_{t \to \, - 1} \frac{{2 - \sqrt {{t^2} + 3} }}{{t + 1}}\).
   Solution
3. For the function \(\displaystyle g\left( \theta\right) = \frac{{\sin \left( {7\theta } \right)}}{\theta }\) answer each of the following questions.
   1. Evaluate the function at the following values of \(\theta \) compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations.
      1. 0.5
      2. 0.1
      3. 0.01
      4. 0.001
      5. 0.0001
      1. -0.5
      2. -0.1
      3. -0.01
      4. -0.001
      5. -0.0001
   2. Use the information from (a) to estimate the value of \(\displaystyle \mathop {\lim }\limits_{\theta\to \,0} \frac{{\sin \left( {7\theta } \right)}}{\theta }\).
   Solution
4. Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\) and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\). If any of the quantities do not exist clearly explain why.
   1. \(a = - 3\)
   2. \(a = - 1\)
   3. \(a = 2\)
   4. \(a = 4\)
   Solution
   
5. Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\) and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\). If any of the quantities do not exist clearly explain why.
   1. \(a = - 8\)
   2. \(a = - 2\)
   3. \(a = 6\)
   4. \(a = 10\)
   Solution
   
6. Below is the graph of \(f\left( x \right)\). For each of the given points determine the value of \(f\left( a \right)\) and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\). If any of the quantities do not exist clearly explain why.
   1. \(a = - 2\)
   2. \(a = - 1\)
   3. \(a = 1\)
   4. \(a = 3\)
   Solution