// Numbas version: exam_results_page_options {"name": "Maria's copy of Limits: L'Hospital's rule: Indeterminate form 0*infinity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "

As {choice2[0]}, it appears {choice2[2]} approaches {choice2[1]}, but it isn't clear what this means. This is known as an indeterminate form of type {choice2[1]}. We rewrite this product as a quotient {choice2[4]} so that it is an indeterminate form of type {choice2[3]}. L'Hospital's rule says that if we have such an indeterminate form we can differentiate the numerator and denominator separately and take the limit of the quotient. That is, if $f$ and $g$ are differentiable on an open interval containing $a$, $g'(x)\\ne 0$ on that interval (except possibly at $x=a$), $\\lim_{x\\rightarrow a}f(x)$ and $\\lim_{x\\rightarrow a}g(x)$ both equal $\\infty$ or both equal $0$ and $\\lim_{x\\rightarrow a}\\frac{f'(x)}{g'(x)}$ exists (or equals $\\pm\\infty)$ then:

\n

\\[\\lim_{x\\rightarrow a} \\frac{f(x)}{g(x)}=\\lim_{x\\rightarrow a}\\frac{f'(x)}{g'(x)}.\\]

\n

(If this is still an indeterminate form we can (of course) repeat the procedure)

\n

\n

So by applying L'Hospital's rule we now need to determine what {choice2[5]} approaches as {choice2[0]}. But this is again an indeterminate form so we repeatedly apply L'Hospital's rule until it is not an indeterminate form and we arrive at needing to determine what {choice2[6]} approaches as {choice2[0]}. Now that this is no longer an indeterminate form, it should be clear that the limit is {choice2[-1]}.

\n

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As {choice2[0]}, {choice2[2]} approaches [[0]] 

\n

Note: infinity is simply entered by typing infinity

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using a list to keep track of important things

\n

what x approaches, indeterminate form type, what y is, the first application of L'Hospital's, (if needed) the final application of L'Hopital's, what y approaches. 

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using a list to keep track of important things

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what x approaches, indeterminate form type, what y is, the first application of L'Hospital's, (if needed) the final application of L'Hopital's, what y approaches. 

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using a list to keep track of important things

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what x approaches, indeterminate form type, what y is, the first application of L'Hospital's, (if needed) the final application of L'Hopital's, what y approaches, latex display of what y approaches.

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using a list to keep track of important things

\n

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using a list to keep track of important things

\n

what x approaches, indeterminate form type, what y is, the first application of L'Hospital's, (if needed) the final application of L'Hopital's, what y approaches, latex display of what y approaches.

"}, "pL": {"templateType": "anything", "definition": "shuffle(2..12)[0..5]", "name": "pL", "group": "Ungrouped variables", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Just what the title says, I guess.

"}, "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "rulesets": {}, "statement": "

This question is about limits of indeterminate forms.

", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}