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  ### Section 4-10 : L'Hospital's Rule and Indeterminate Forms
  
  Use Lâ€™Hospitalâ€™s Rule to evaluate each of the following limits.
  
  1. $$\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{{x^3} - 7{x^2} + 10x}}{{{x^2} + x - 6}}$$ Solution
  2. $$\displaystyle \mathop {\lim }\limits_{w \to \, - 4} \frac{{\sin \left( {\pi w} \right)}}{{{w^2} - 16}}$$ Solution
  3. $$\displaystyle \mathop {\lim }\limits_{t \to \infty } \frac{{\ln \left( {3t} \right)}}{{{t^2}}}$$ Solution
  4. $$\displaystyle \mathop {\lim }\limits_{z \to 0} \frac{{\sin \left( {2z} \right) + 7{z^2} - 2z}}{{{z^2}{{\left( {z + 1} \right)}^2}}}$$ Solution
  5. $$\displaystyle \mathop {\lim }\limits_{x \to \, - \infty } \frac{{{x^2}}}{{{{\bf{e}}^{1 - \,x}}}}$$ Solution
  6. $$\displaystyle \mathop {\lim }\limits_{z \to \infty } \frac{{{z^2} + {{\bf{e}}^{4\,z}}}}{{2z - {{\bf{e}}^{\,z}}}}$$ Solution
  7. $$\mathop {\lim }\limits_{t \to \infty } \left[ {t\ln \left( {1 + \displaystyle \frac{3}{t}} \right)} \right]$$ Solution
  8. $$\mathop {\lim }\limits_{w \to {0^ + }} \left[ {{w^2}\ln \left( {4{w^2}} \right)} \right]$$ Solution
  9. $$\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\displaystyle \frac{\pi }{2}x} \right)} \right]$$ Solution
  10. $$\mathop {\lim }\limits_{y \to {0^ + }} {\left[ {\cos \left( {2y} \right)} \right]^{{}^{1}/{}_{{{y^{\,2}}}}}}$$ Solution
  11. $$\mathop {\lim }\limits_{x \to \infty } {\left[ {{{\bf{e}}^x} + x} \right]^{{}^{1}/{}_{x}}}$$ Solution
explanation: The document is a section from Paul's Online Notes, specifically from the Calculus I course, focusing on L'Hospital's Rule and Indeterminate Forms. It contains a list of 11 limits that need to be evaluated using L'Hospital's Rule. Each limit is presented as a separate problem, and the document indicates that solutions are available for each problem.
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distilabel_metadata.content: Paul's Online Notes
  Home / Calculus I / Applications of Derivatives / L'Hospital's Rule and Indeterminate Forms
  Show Mobile Notice Show All Notes Hide All Notes
  Mobile Notice
  You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
  
  ### Section 4-10 : L'Hospital's Rule and Indeterminate Forms
  
  Use Lâ€™Hospitalâ€™s Rule to evaluate each of the following limits.
  
  1. $$\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{{x^3} - 7{x^2} + 10x}}{{{x^2} + x - 6}}$$ Solution
  2. $$\displaystyle \mathop {\lim }\limits_{w \to \, - 4} \frac{{\sin \left( {\pi w} \right)}}{{{w^2} - 16}}$$ Solution
  3. $$\displaystyle \mathop {\lim }\limits_{t \to \infty } \frac{{\ln \left( {3t} \right)}}{{{t^2}}}$$ Solution
  4. $$\displaystyle \mathop {\lim }\limits_{z \to 0} \frac{{\sin \left( {2z} \right) + 7{z^2} - 2z}}{{{z^2}{{\left( {z + 1} \right)}^2}}}$$ Solution
  5. $$\displaystyle \mathop {\lim }\limits_{x \to \, - \infty } \frac{{{x^2}}}{{{{\bf{e}}^{1 - \,x}}}}$$ Solution
  6. $$\displaystyle \mathop {\lim }\limits_{z \to \infty } \frac{{{z^2} + {{\bf{e}}^{4\,z}}}}{{2z - {{\bf{e}}^{\,z}}}}$$ Solution
  7. $$\mathop {\lim }\limits_{t \to \infty } \left[ {t\ln \left( {1 + \displaystyle \frac{3}{t}} \right)} \right]$$ Solution
  8. $$\mathop {\lim }\limits_{w \to {0^ + }} \left[ {{w^2}\ln \left( {4{w^2}} \right)} \right]$$ Solution
  9. $$\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\displaystyle \frac{\pi }{2}x} \right)} \right]$$ Solution
  10. $$\mathop {\lim }\limits_{y \to {0^ + }} {\left[ {\cos \left( {2y} \right)} \right]^{{}^{1}/{}_{{{y^{\,2}}}}}}$$ Solution
  11. $$\mathop {\lim }\limits_{x \to \infty } {\left[ {{{\bf{e}}^x} + x} \right]^{{}^{1}/{}_{x}}}$$ Solution
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a section from Paul's Online Notes, specifically from the Calculus I course, focusing on L'Hospital's Rule and Indeterminate Forms. It contains a list of 11 limits that need to be evaluated using L'Hospital's Rule. Each limit is presented as a separate problem, and the document indicates that solutions are available for each problem.
  
  Label: multiple-question