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

As {choice2}, it appears {choice2} approaches {choice2}, but it isn't clear what this means. This is known as an indeterminate form of type {choice2}. 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 $x=a$), $\\lim_{x\\rightarrow a}f(x)=\\infty$, $\\lim_{x\\rightarrow a}g(x)=\\infty$ 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} approaches as {choice2}. 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} approaches as {choice2}. Now that this is no longer an indeterminate form, it should be clear that the limit is $\\var{choice2[-1]}$.

\n

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Just what the title says, I guess.

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using a list to keep track of important things

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using a list to keep track of important things

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"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "type": "numberentry", "minValue": "choice2[-1]", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "mustBeReducedPC": 0, "variableReplacements": [], "maxValue": "choice2[-1]"}], "showFeedbackIcon": true, "marks": 0, "prompt": "

As {choice2}, {choice2} approaches []

\n

Note: infinity is simply entered by typing infinity

\n

", "scripts": {}, "variableReplacements": []}], "statement": "

This question is about limits of indeterminate forms.

", "tags": [], "name": "Sid's copy of Limits: L'Hospital's rule: Indeterminate form infinity/infinity", "variable_groups": [], "rulesets": {}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Sid Grover", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3442/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Sid Grover", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3442/"}]}