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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
The text passage is a section from a calculus course, specifically dealing with L'Hospital's Rule and Indeterminate Forms. It lists several limit problems that require the application of L'Hospital's Rule to solve, with a note that solutions are provided for each problem.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Paul's Online Notes
Home / Calculus I / Applications of Derivatives / L'Hospital's Rule and Indeterminate Forms
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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
user
The text passage is a section from a calculus course, specifically dealing with L'Hospital's Rule and Indeterminate Forms. It lists several limit problems that require the application of L'Hospital's Rule to solve, with a note that solutions are provided for each problem.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS