// Numbas version: finer_feedback_settings
{"name": "Limits: L'Hospital's rule: Indeterminate form infinity/infinity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"prompt": "<p>As {choice2[0]}, {choice2[2]} approaches&nbsp;[[0]]&nbsp;</p>\n<p><strong>Note:</strong>&nbsp;infinity is simply entered by typing&nbsp;<strong>infinity</strong></p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "gaps": [{"minValue": "choice2[-1]", "mustBeReduced": false, "mustBeReducedPC": 0, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "choice2[-1]", "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "allowFractions": false, "variableReplacements": []}], "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "marks": 0, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "statement": "<p>This question is about limits of indeterminate forms.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "advice": "<p>As {choice2[0]},&nbsp;it appears {choice2[2]}&nbsp;approaches {choice2[1]}, but&nbsp;it isn't clear what this means. This is known as an indeterminate form of type {choice2[1]}.&nbsp;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:</p>\n<p>\\[\\lim_{x\\rightarrow a} \\frac{f(x)}{g(x)}=\\lim_{x\\rightarrow a}\\frac{f'(x)}{g'(x)}.\\]</p>\n<p>(If this is still an indeterminate form we can (of course) repeat the procedure)</p>\n<p></p>\n<p>So by applying L'Hospital's rule we now need to determine what {choice2[3]} approaches as&nbsp;{choice2[0]}. <span data-jme-visible=\"len(choice2)=6\">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[4]} approaches as&nbsp;{choice2[0]}.</span>&nbsp;Now that this is no longer an indeterminate form, it should be clear that&nbsp;the limit is $\\var{choice2[-1]}$.</p>\n<p></p>", "metadata": {"description": "<p>Just what the title says, I guess.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "name": "Limits: L'Hospital's rule: Indeterminate form infinity/infinity", "rulesets": {}, "ungrouped_variables": ["pL", "pa", "pb", "pc", "pd", "pe", "nL", "na", "nb", "nc", "nd", "ne", "log_power", "power_log", "exp_power", "power_exp", "exp_log", "log_exp", "log_root", "root_log", "choice1", "choice2"], "extensions": [], "tags": [], "variable_groups": [], "variables": {"power_log": {"definition": "[\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pd}x^{pb}+{nd})/({pc}ln({pa}x+{nb}))}\\$','\\$\\\\simplify{({pd*pb}x^{pb-1})/({pc*pa}/({pa}x+{nb}))=(({pa}x+{nb})*({pd*pb}x^{pb-1}))/({pc*pa})}\\$',infinity], \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pd}x^{pb}+{nd})/({pc}ln({pa}x+{nb}))}\\$','\\$\\\\simplify{({-pd*pb}x^{pb-1})/({pc*pa}/({pa}x+{nb}))=(({pa}x+{nb})*({-pd*pb}x^{pb-1}))/({pc*pa})}\\$',-infinity], \n\n\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({-pd}x^{pb}+{nd})/({pc}ln({-pa}x+{nb}))}\\$','\\$\\\\simplify{({-pd*pb}x^{pb-1})/({-pc*pa}/({-pa}x+{nb}))=(({-pa}x+{nb})*({-pd*pb}x^{pb-1}))/({-pc*pa})}\\$',if(mod(pb,2)=0,-infinity,infinity)],\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({pd}x^{pb}+{nd})/({pc}ln({-pa}x+{nb}))}\\$','\\$\\\\simplify{({pd*pb}x^{pb-1})/({-pc*pa}/({-pa}x+{nb}))=(({-pa}x+{nb})*({pd*pb}x^{pb-1}))/({-pc*pa})}\\$',if(mod(pb,2)=0,infinity,-infinity)]\n\n\n]\n", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "power_log", "templateType": "anything", "group": "Ungrouped variables"}, "nL": {"definition": "shuffle(-12..12 except 0)[0..5]", "description": "", "name": "nL", "templateType": "anything", "group": "Ungrouped variables"}, "pa": {"definition": "pl[0]", "description": "", "name": "pa", "templateType": "anything", "group": "Ungrouped variables"}, "nd": {"definition": "nl[3]", "description": "", "name": "nd", "templateType": "anything", "group": "Ungrouped variables"}, "nc": {"definition": "nl[2]", "description": "", "name": "nc", "templateType": "anything", "group": "Ungrouped variables"}, "log_exp": {"definition": "[\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pd}ln({pb}x+{nb}))/({pc}e^({pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({pd*pb}/({pb}x+{nb}))/({pc}*{pa}*e^({pa}x+{nb}))={pd*pb}/({pc}*{pa}*e^({pa}x+{nb})*({pb}x+{nb}))}\\$',0], \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pd}ln({pb}x+{nb}))/({pc}e^({pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({-pd*pb}/({pb}x+{nb}))/({pc}*{pa}*e^({pa}x+{nb}))={-pd*pb}/({pc}*{pa}*e^({pa}x+{nb})*({pb}x+{nb}))}\\$',0], \n\n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$', '\\$\\\\simplify{({pd}ln({-pb}x+{nb}))/({pc}e^({-pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({pd*-pb}/({-pb}x+{nb}))/({pc}*{-pa}*e^({-pa}x+{nb}))={pd*-pb}/({pc}*{-pa}*e^({-pa}x+{nb})*({-pb}x+{nb}))}\\$',0],\n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$', '\\$\\\\simplify{({-pd}ln({-pb}x+{nb}))/({pc}e^({-pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({pd*pb}/({-pb}x+{nb}))/({pc}*{-pa}*e^({-pa}x+{nb}))={pd*pb}/({pc}*{-pa}*e^({-pa}x+{nb})*({-pb}x+{nb}))}\\$',0]\n]", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "log_exp", "templateType": "anything", "group": "Ungrouped variables"}, "pc": {"definition": "pl[2]", "description": "", "name": "pc", "templateType": "anything", "group": "Ungrouped variables"}, "ne": {"definition": "nl[4]", "description": "", "name": "ne", "templateType": "anything", "group": "Ungrouped variables"}, "exp_power": {"definition": "[\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pc}e^({pa}x+{nb})+{nc})/({pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({pc}*{pa}*e^({pa}x+{nb}))/({pd}*{pb}x^{pb-1})}\\$','\\$\\\\simplify[!othernumbers]{({pc}*{pa}^{pb}*e^({pa}x+{nb}))/({pd}*{pb}!)}\\$',infinity], \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pc}e^({pa}x+{nb})+{nc})/({pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({-pc}*{pa}*e^({pa}x+{nb}))/({pd}*{pb}x^{pb-1})}\\$','\\$\\\\simplify[!othernumbers]{({-pc}*{pa}^{pb}*e^({pa}x+{nb}))/({pd}*{pb}!)}\\$',-infinity], \n\n\n\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{-\\\\infty}\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({pc}e^({-pa}x+{nb})+{nc})/({-pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({pc}*{-pa}e^({-pa}x+{nb}))/({-pd}*{pb}x^{pb-1})}\\$','\\$\\\\simplify[!othernumbers]{({pc}*{-pa}^{pb}*e^({-pa}x+{nb}))/({-pd}*{pb}!)}\\$',if(mod(pb,2)=0,-infinity,infinity)],\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\frac{\\\\infty}{-\\\\infty}\\$'), '\\$\\\\simplify{({-pc}e^({-pa}x+{nb})+{nc})/({-pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({pc}*{pa}e^({-pa}x+{nb}))/({-pd}*{pb}x^{pb-1})}\\$','\\$\\\\simplify[!othernumbers]{({pc}*{pa}^{pb}*e^({-pa}x+{nb}))/({-pd}*{pb}!)}\\$',if(mod(pb,2)=0,infinity,-infinity)]\n]\n", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "exp_power", "templateType": "anything", "group": "Ungrouped variables"}, "pe": {"definition": "pl[4]", "description": "", "name": "pe", "templateType": "anything", "group": "Ungrouped variables"}, "pd": {"definition": "pl[3]", "description": "", "name": "pd", "templateType": "anything", "group": "Ungrouped variables"}, "exp_log": {"definition": "[\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pd}e^({pa}x+{nb})+{nc})/({pc}ln({pb}x+{nb}))}\\$','\\$\\\\simplify{({pd}*{pa}*e^({pa}x+{nb}))/({pc*pb}/({pb}x+{nb}))=({pd}*{pa}*e^({pa}x+{nb})*({pb}x+{nb}))/{pc*pb}}\\$',infinity], \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pd}e^({pa}x+{nb})+{nc})/({pc}ln({pb}x+{nb}))}\\$','\\$\\\\simplify{({-pd}*{pa}*e^({pa}x+{nb}))/({pc*pb}/({pb}x+{nb}))=({-pd}*{pa}*e^({pa}x+{nb})*({pb}x+{nb}))/{pc*pb}}\\$',-infinity], \n\n\n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$', '\\$\\\\simplify{({pd}e^({-pa}x+{nb})+{nc})/({pc}ln({-pb}x+{nb}))}\\$','\\$\\\\simplify{({pd}*{-pa}*e^({-pa}x+{nb}))/({pc*-pb}/({-pb}x+{nb}))=({pd}*{-pa}*e^({-pa}x+{nb})*({-pb}x+{nb}))/{pc*-pb}}\\$',infinity],\n  ['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$', '\\$\\\\simplify{({-pd}e^({-pa}x+{nb})+{nc})/({pc}ln({-pb}x+{nb}))}\\$','\\$\\\\simplify{({-pd}*{-pa}*e^({-pa}x+{nb}))/({pc*-pb}/({-pb}x+{nb}))=({pd}*{pa}*e^({-pa}x+{nb})*({-pb}x+{nb}))/{pc*-pb}}\\$',-infinity]\n]", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "exp_log", "templateType": "anything", "group": "Ungrouped variables"}, "choice1": {"definition": "random(0..7)", "description": "", "name": "choice1", "templateType": "anything", "group": "Ungrouped variables"}, "nb": {"definition": "nl[1]", "description": "", "name": "nb", "templateType": "anything", "group": "Ungrouped variables"}, "root_log": {"definition": "[\n\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{(root({pa}x+{nb},{pb}))/({pc}ln({pa}x+{nb}))}\\$','\\$\\\\simplify{({pa}/{pb}*(({pa}x+{nb})^({1-pb}/{pb})))/({pc*pa}/({pa}x+{nb}))=(({pa}x+{nb})^(1/{pb}))/({pc*pb})}\\$',infinity], \n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{(root({-pa}x+{nb},{pb}))/({pc}ln({-pa}x+{nb}))}\\$','\\$\\\\simplify{({-pa}/{pb}*(({-pa}x+{nb})^({1-pb}/{pb})))/({-pc*pa}/({-pa}x+{nb}))=(({-pa}x+{nb})^(1/{pb}))/({pc*pb})}\\$',infinity], \n  \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{(root({pa}x+{nb},{pb}))/({-pc}ln({pa}x+{nb}))}\\$','\\$\\\\simplify{({pa}/{pb}*(({pa}x+{nb})^({1-pb}/{pb})))/({-pc*pa}/({pa}x+{nb}))=(({pa}x+{nb})^(1/{pb}))/({-pc*pb})}\\$',-infinity], \n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{(root({-pa}x+{nb},{pb}))/({-pc}ln({-pa}x+{nb}))}\\$','\\$\\\\simplify{({-pa}/{pb}*(({-pa}x+{nb})^({1-pb}/{pb})))/({pc*pa}/({-pa}x+{nb}))=(({-pa}x+{nb})^(1/{pb}))/({-pc*pb})}\\$',-infinity]\n\n]", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "root_log", "templateType": "anything", "group": "Ungrouped variables"}, "choice2": {"definition": "switch(choice1=0,random(log_power),choice1=1,random(power_log),choice1=2,random(exp_power),choice1=3,random(power_exp),choice1=4,random(exp_log),choice1=5,random(log_exp),choice1=6,random(log_root),choice1=7,random(root_log),'')", "description": "", "name": "choice2", "templateType": "anything", "group": "Ungrouped variables"}, "pb": {"definition": "pl[1]", "description": "", "name": "pb", "templateType": "anything", "group": "Ungrouped variables"}, "power_exp": {"definition": "[\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pd}x^{pb}+{nd})/({pc}e^({pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({pd}*{pb}x^{pb-1})/({pc}*{pa}*e^({pa}x+{nb}))}\\$','\\$\\\\simplify[!othernumbers]{({pd}*{pb}!)/({pc}*{pa}^{pb}*e^({pa}x+{nb}))}\\$',0], \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pd}x^{pb}+{nd})/({pc}e^({pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({-pd}*{pb}x^{pb-1})/({pc}*{pa}*e^({pa}x+{nb}))}\\$','\\$\\\\simplify[!othernumbers]{({-pd}*{pb}!)/({pc}*{pa}^{pb}*e^({pa}x+{nb}))}\\$',0], \n\n\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({-pd}x^{pb}+{nd})/({pc}e^({-pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({-pd}*{pb}x^{pb-1})/({pc}*{-pa}e^({-pa}x+{nb}))}\\$','\\$\\\\simplify[!othernumbers]{({-pd}*{pb}!)/({pc}*{-pa}^{pb}*e^({-pa}x+{nb}))}\\$',0],\n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({pd}x^{pb}+{nd})/({pc}e^({-pa}x+{nb})+{nc})}\\$','\\$\\\\simplify{({pd}*{pb}x^{pb-1})/({pc}*{-pa}e^({-pa}x+{nb}))}\\$','\\$\\\\simplify[!othernumbers]{({pd}*{pb}!)/({pc}*{-pa}^{pb}*e^({-pa}x+{nb}))}\\$',0]\n]\n", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "power_exp", "templateType": "anything", "group": "Ungrouped variables"}, "na": {"definition": "nl[0]", "description": "", "name": "na", "templateType": "anything", "group": "Ungrouped variables"}, "log_power": {"definition": "[\n\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pc}ln({pa}x+{nb}))/({pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({pc*pa}/({pa}x+{nb}))/({pd*pb}x^{pb-1})=({pc*pa})/(({pa}x+{nb})*({pd*pb}x^{pb-1}))}\\$',0], \n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{-\\\\infty}\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({pc}ln({-pa}x+{nb}))/({-pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({-pc*pa}/({-pa}x+{nb}))/({-pd*pb}x^{pb-1})=({-pc*pa})/(({-pa}x+{nb})*({-pd*pb}x^{pb-1}))}\\$',0]\n,  \n ['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pc}ln({pa}x+{nb}))/({pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({-pc*pa}/({pa}x+{nb}))/({pd*pb}x^{pb-1})=({-pc*pa})/(({pa}x+{nb})*({pd*pb}x^{pb-1}))}\\$',0], \n['\\$x\\\\rightarrow-\\\\infty\\$',if(mod(pb,2)=0,'\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$'), '\\$\\\\simplify{({-pc}ln({-pa}x+{nb}))/({-pd}x^{pb}+{nd})}\\$','\\$\\\\simplify{({pc*pa}/({-pa}x+{nb}))/({-pd*pb}x^{pb-1})=({pc*pa})/(({-pa}x+{nb})*({-pd*pb}x^{pb-1}))}\\$',0]\n]", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "log_power", "templateType": "anything", "group": "Ungrouped variables"}, "log_root": {"definition": "[\n\n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pc}ln({pa}x+{nb}))/(root({pa}x+{nb},{pb}))}\\$','\\$\\\\simplify{({pc*pa}/({pa}x+{nb}))/({pa}/{pb}*(({pa}x+{nb})^({1-pb}/{pb})))=({pc*pb})/(({pa}x+{nb})^(1/{pb}))}\\$',0], \n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({pc}ln({-pa}x+{nb}))/(root({-pa}x+{nb},{pb}))}\\$','\\$\\\\simplify{({-pc*pa}/({-pa}x+{nb}))/({-pa}/{pb}*(({-pa}x+{nb})^({1-pb}/{pb})))=({pc*pb})/(({-pa}x+{nb})^(1/{pb}))}\\$',0], \n  \n['\\$x\\\\rightarrow\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pc}ln({pa}x+{nb}))/(root({pa}x+{nb},{pb}))}\\$','\\$\\\\simplify{({-pc*pa}/({pa}x+{nb}))/({pa}/{pb}*(({pa}x+{nb})^({1-pb}/{pb})))=({-pc*pb})/(({pa}x+{nb})^(1/{pb}))}\\$',0], \n['\\$x\\\\rightarrow-\\\\infty\\$','\\$\\\\frac{-\\\\infty}{\\\\infty}\\$','\\$\\\\simplify{({-pc}ln({-pa}x+{nb}))/(root({-pa}x+{nb},{pb}))}\\$','\\$\\\\simplify{({pc*pa}/({-pa}x+{nb}))/({-pa}/{pb}*(({-pa}x+{nb})^({1-pb}/{pb})))=({-pc*pb})/(({-pa}x+{nb})^(1/{pb}))}\\$',0]\n\n]", "description": "<p>using a list to keep track of important things</p>\n<p>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.&nbsp;</p>", "name": "log_root", "templateType": "anything", "group": "Ungrouped variables"}, "pL": {"definition": "shuffle(1..12)[0..5]", "description": "", "name": "pL", "templateType": "anything", "group": "Ungrouped variables"}}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}