// Numbas version: exam_results_page_options
{"name": "Maria's copy of Limits: L'Hospital's rule: Indeterminate form 0/0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "parts": [{"scripts": {}, "type": "gapfill", "gaps": [{"checkingtype": "absdiff", "type": "jme", "showCorrectAnswer": false, "vsetrangepoints": 5, "showFeedbackIcon": true, "expectedvariablenames": [], "variableReplacements": [], "answer": "{{choice2}[-2]}", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "showpreview": true, "checkvariablenames": false, "checkingaccuracy": "0.0000000001", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "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>", "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "extensions": [], "tags": [], "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 $x=a$), $\\lim_{x\\rightarrow a}f(x)=0$, $\\lim_{x\\rightarrow a}g(x)=0$ 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)=7\">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 {choice2[-1]}.</p>\n<p></p>", "name": "Maria's copy of Limits: L'Hospital's rule: Indeterminate form 0/0", "statement": "<p>This question is about limits of indeterminate forms.</p>", "variable_groups": [], "preamble": {"css": "", "js": ""}, "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Just what the title says, I guess.</p>"}, "variables": {"pc": {"name": "pc", "definition": "pl[2]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "na": {"name": "na", "definition": "nl[0]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "nb": {"name": "nb", "definition": "nl[1]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "pe": {"name": "pe", "definition": "pl[4]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "poly_sin": {"name": "poly_sin", "definition": 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0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{x^2/({nc}sin(x))}\\$','\\$\\\\simplify{(2x)/({nc}cos(x))}\\$',0,'\\$ 0\\$'],\n  \n['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x-{na})^{pa}/({nc}sin(x-{na}))}\\$','\\$\\\\simplify{({pa}(x-{na})^{pa-1})/({nc}cos(x-{na}))}\\$',0,'\\$ 0\\$']\n\n]", "templateType": "anything", "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>", "group": "Ungrouped variables"}, "pd": {"name": "pd", "definition": "pl[3]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "alt1": {"name": "alt1", "definition": "if(mod(pa,4)=0 or mod(pa,4)=1,1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "log_cos": {"name": "log_cos", "definition": 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\n['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x/{na}))/({pa}cos(x-{na})-{pa})}\\$','\\$\\\\simplify{({nb})/(x*({-pa}sin(x-{na})))}\\$',-nb/(na)*infinity,'\\$\\\\simplify{{-nb/(na)*infinity}}\\$'], \n['\\$x\\\\rightarrow\\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x/{na}))/({pa}cos(x-{na})-{pa})}\\$','\\$\\\\simplify{({nb})/(x*({-pa}sin(x-{na})))}\\$',nb/(na)*infinity,'\\$\\\\simplify{{nb/(na)*infinity}}\\$'],\n \n['\\$x\\\\rightarrow 1\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x))/({pa}cos(pi/2*x))}\\$','\\$\\\\simplify[fractionNumbers,simplifyFractions]{({2*nb})/((x)*({-pa}*pi*sin(pi/2*x)))}\\$',2*nb/(-pa*pi),'\\$\\\\simplify[fractionNumbers,simplifyFractions,unitFactor]{{-2*nb}/({pa}*{pi})}\\$'],  \n  \n['\\$x\\\\rightarrow 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"poly_cos": {"name": "poly_cos", "definition": "[\n\n['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{((x-{na})*(x-{nb}))/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{(2x-{na+nb})/({-nc}sin(x-{na}))}\\$',(na-nb)/(-nc)*infinity,'\\$\\\\simplify{{(na-nb)/(-nc)*infinity}}\\$'],\n['\\$x\\\\rightarrow\\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{((x-{na})*(x-{nb}))/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{(2x-{na+nb})/({-nc}sin(x-{na}))}\\$',(na-nb)/(nc)*infinity,'\\$\\\\simplify{{(na-nb)/(nc)*infinity}}\\$'],   \n['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x^2-{na+nb}x+{na*nb})/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{(2x-{na+nb})/({-nc}sin(x-{na}))}\\$',(na-nb)/(-nc)*infinity,'\\$\\\\simplify{{(na-nb)/(-nc)*infinity}}\\$'],\n['\\$x\\\\rightarrow\\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x^2-{na+nb}x+{na*nb})/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{(2x-{na+nb})/({-nc}sin(x-{na}))}\\$',(na-nb)/(nc)*infinity,'\\$\\\\simplify{{(na-nb)/(nc)*infinity}}\\$'],   \n  \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x-{na})^2/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{(2(x-{na}))/({-nc}sin(x-{na}))}\\$','\\$\\\\simplify{(2)/({-nc}cos(x-{na}))}\\$',-2/nc,'\\$\\\\simplify[fractionNumbers]{{-2/nc}}\\$'],\n\n['\\$x\\\\rightarrow 0^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x^2-{nb}x)/({nc}cos(x)-{nc})}\\$','\\$\\\\simplify{(2x-{nb})/({-nc}sin(x))}\\$',(nb/nc)*infinity,'\\$\\\\simplify{{(nb/nc)*infinity}}\\$'], \n['\\$x\\\\rightarrow 0^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x^2-{nb}x)/({nc}cos(x)-{nc})}\\$','\\$\\\\simplify{(2x-{nb})/({-nc}sin(x))}\\$',(-nb/nc)*infinity,'\\$\\\\simplify{{(-nb/nc)*infinity}}\\$'], \n\n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x^2)/({nc}cos(x)-{nc})}\\$','\\$\\\\simplify{(2x)/({-nc}sin(x))}\\$','\\$\\\\simplify{(2)/({-nc}cos(x))}\\$',(-2/nc),'\\$\\\\simplify[fractionNumbers]{{-2/nc}}\\$'], \n     \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{(x-{na})^{pa}/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{({pa}(x-{na})^{pa-1})/({-nc}sin(x-{na}))}\\$','\\$\\\\simplify[unitPower,zeroPower]{({pa*(pa-1)}(x-{na})^{pa-2})/({-nc}cos(x-{na}))}\\$',if(pa=2,-2/nc,0),if(pa=2,'\\$\\\\simplify[fractionNumbers]{{-2/nc}}\\$','\\$ 0\\$')]\n\n]", "templateType": "anything", "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>", "group": "Ungrouped variables"}, "choice1": {"name": "choice1", "definition": "random(sin_poly,cos_poly,poly_sin,poly_cos,trig_trig,log_sin,log_cos,log_tan,sin_log,cos_log,tan_log)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "pL": {"name": "pL", "definition": "shuffle(2..12)[0..5]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "ne": {"name": "ne", "definition": "nl[4]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "sin_poly": {"name": "sin_poly", "definition": "[\n\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x-{na}))/((x-{na})*(x-{nb}))}\\$','\\$\\\\simplify{({nc}cos(x-{na}))/(2x-{na+nb})}\\$',nc/(na-nb),'\\$\\\\simplify[fractionNumbers]{{nc/(na-nb)}}\\$'], \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x-{na}))/(x^2-{na+nb}x+{na*nb})}\\$','\\$\\\\simplify{({nc}cos(x-{na}))/(2x-{na+nb})}\\$',nc/(na-nb),'\\$\\\\simplify[fractionNumbers]{{nc/(na-nb)}}\\$'],\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x-{na}))/(x-{na})^2}\\$','\\$\\\\simplify{({nc}cos(x-{na}))/(2(x-{na}))}\\$',0,'\\$0\\$'],\n\n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x))/(x^2-{nb}x)}\\$','\\$\\\\simplify{({nc}cos(x))/(2x-{nb})}\\$',-nc/nb,'\\$\\\\simplify[fractionNumbers]{{-nc/nb}}\\$'], \n['\\$x\\\\rightarrow 0^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x))/x^2}\\$','\\$\\\\simplify{({nc}cos(x))/(2x)}\\$',nc*infinity,'\\$\\\\simplify{{nc*infinity}}\\$'],\n['\\$x\\\\rightarrow 0^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x))/x^2}\\$','\\$\\\\simplify{({nc}cos(x))/(2x)}\\$',-nc*infinity,'\\$\\\\simplify{-{nc*infinity}}\\$'],\n  \n['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x-{na}))/(x-{na})^{pa}}\\$','\\$\\\\simplify{({nc}cos(x-{na}))/({pa}(x-{na})^{pa-1})}\\$',nc*infinity,'\\$\\\\simplify{{nc*infinity}}\\$'],\n['\\$x\\\\rightarrow\\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}sin(x-{na}))/(x-{na})^{pa}}\\$','\\$\\\\simplify{({nc}cos(x-{na}))/({pa}(x-{na})^{pa-1})}\\$',if(mod(pa,2)=0,-nc*infinity,nc*infinity),if(mod(pa,2)=0,'\\$\\\\simplify{{-nc*infinity}}\\$','\\$\\\\simplify{{nc*infinity}}\\$')]  \n\n]", "templateType": "anything", "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, latex display of what y approaches.</p>", "group": "Ungrouped variables"}, "choice2": {"name": "choice2", "definition": "random(choice1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "cos_log": {"name": "cos_log", "definition": "[\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}cos(x-{na})-{pa})/({nb}ln(x-{na-1}))}\\$','\\$\\\\simplify{((x-{na-1})*({-pa}sin(x-{na})))/({nb})}\\$',0,'\\$ 0\\$'], \n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}cos(x)-{pa})/({nb}ln(x+1))}\\$','\\$\\\\simplify{((x+1)*({-pa}sin(x)))/({nb})}\\$',0,'\\$ 0 \\$'], \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}cos(x-{na})-{pa})/({nb}ln(x/{na}))}\\$','\\$\\\\simplify{(x*({-pa}sin(x-{na})))/({nb})}\\$',0,'\\$ 0 \\$'], \n['\\$x\\\\rightarrow 1\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}cos(pi/2*x))/({nb}ln(x))}\\$','\\$\\\\simplify{((x)*({-pa}*pi*sin(pi/2*x)))/({2*nb})}\\$',(-pi*pa)/(2*nb),'\\$\\\\simplify[fractionNumbers,simplifyFractions,unitFactor,unitDenominator]{({-pa}*{pi})/{2*nb}}\\$'],\n['\\$x\\\\rightarrow 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\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}cos(x-{na})-{nc})/(x-{na})^2}\\$','\\$\\\\simplify{({-nc}sin(x-{na}))/(2(x-{na}))}\\$','\\$\\\\simplify{({-nc}cos(x-{na}))/(2)}\\$',-nc/2,'\\$\\\\simplify[fractionNumbers]{{-nc/2}}\\$'],\n\n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}cos(x)-{nc})/(x^2-{nb}x)}\\$','\\$\\\\simplify{({-nc}sin(x))/(2x-{nb})}\\$',0,'\\$0\\$'], \n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}cos(x)-{nc})/x^2}\\$','\\$\\\\simplify{({-nc}sin(x))/(2x)}\\$','\\$\\\\simplify{({-nc}cos(x))/(2)}\\$',-nc/2,'\\$\\\\simplify[fractionNumbers]{{-nc/2}}\\$'],\n  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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.</p>", "group": "Ungrouped variables"}, "log_sin": {"name": "log_sin", "definition": "[\n\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x-{na-1}))/({pa}sin(x-{na}))}\\$','\\$\\\\simplify{({nb})/((x-{na-1})*({pa}cos(x-{na})))}\\$',nb/pa,'\\$\\\\simplify[fractionNumbers]{{nb/pa}}\\$'], \n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x+1))/({pa}sin(x))}\\$','\\$\\\\simplify{({nb})/((x+1)*({pa}cos(x)))}\\$',nb/pa,'\\$\\\\simplify[fractionNumbers]{{nb/pa}}\\$'], \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x/{na}))/({pa}sin(x-{na}))}\\$','\\$\\\\simplify{({nb})/(x*({pa}cos(x-{na})))}\\$',nb/(pa*na),'\\$\\\\simplify[fractionNumbers,simplifyFractions]{{nb}/{pa*na}}\\$'], \n['\\$x\\\\rightarrow 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"[\n\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x-{na-1}))/({pa}tan(x-{na}))}\\$','\\$\\\\simplify{({nb})/((x-{na-1})*({pa}sec(x-{na})^2))=({nb}cos(x-{na})^2)/((x-{na-1})*{pa})}\\$',nb/pa,'\\$\\\\simplify[fractionNumbers]{{nb/pa}}\\$'], \n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x+1))/({pa}tan(x))}\\$','\\$\\\\simplify{({nb})/((x+1)*{pa}*sec(x)^2)=({nb}cos(x)^2)/((x+1)*{pa})}\\$',nb/pa,'\\$\\\\simplify[fractionNumbers]{{nb/pa}}\\$'], \n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x/{na}))/({pa}tan(x-{na}))}\\$','\\$\\\\simplify{({nb})/(x*{pa}*sec(x-{na})^2)=({nb}*cos(x-{na})^2)/(x*{pa})}\\$',nb/(pa*na),'\\$\\\\simplify[fractionNumbers,simplifyFractions]{{nb}/{pa*na}}\\$'], \n['\\$x\\\\rightarrow 1\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x))/({pa}tan(pi*x))}\\$','\\$\\\\simplify{({nb})/((x)*{pa}*pi*sec(pi*x)^2)=({nb}*cos(pi*x)^2)/((x)*{pa}*pi)}\\$',nb/(pi*pa),'\\$\\\\simplify[fractionNumbers,simplifyFractions,unitFactor]{{nb}/({pa}*{pi})}\\$'],\n['\\$x\\\\rightarrow \\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x-{na-1}))/({pa}tan((x-{na})^2))}\\$','\\$\\\\simplify{({nb})/((x-{na-1})*{pa}*2(x-{na})*sec((x-{na})^2)^2)=({nb}*cos((x-{na})^2)^2)/((x-{na-1})*{pa}*2(x-{na}))}\\$',nb*infinity,'\\$\\\\simplify{{nb*infinity}}\\$'],\n['\\$x\\\\rightarrow \\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nb}ln(x-{na-1}))/({pa}tan((x-{na})^2))}\\$','\\$\\\\simplify{({nb})/((x-{na-1})*{pa}*2(x-{na})*sec((x-{na})^2)^2)=({nb}*cos((x-{na})^2)^2)/((x-{na-1})*{pa}*2(x-{na}))}\\$',-nb*infinity,'\\$\\\\simplify{{-nb*infinity}}\\$']\n]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "trig_trig": {"name": "trig_trig", "definition": "[\n\n['\\$x\\\\rightarrow\\\\var{na}\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}cos(x-{na})-{nc})/({pa}sin(x-{na}))}\\$','\\$\\\\simplify{({-nc}sin(x-{na}))/({pa}cos(x-{na}))}\\$',0,'\\$ 0\\$'], \n\n['\\$x\\\\rightarrow 0\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({nc}cos(x)-{nc})/({pa}sin(x))}\\$','\\$\\\\simplify{({-nc}sin(x))/({pa}cos(x))}\\$',0,'\\$0\\$'], \n\n  ['\\$x\\\\rightarrow\\\\var{na}^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}sin(x-{na}))/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{({pa}cos(x-{na}))/({-nc}sin(x-{na}))}\\$',(-pa/nc)*infinity,'\\$ \\\\simplify{{(-pa/nc)*infinity}}\\$'], \n  ['\\$x\\\\rightarrow\\\\var{na}^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}sin(x-{na}))/({nc}cos(x-{na})-{nc})}\\$','\\$\\\\simplify{({pa}cos(x-{na}))/({-nc}sin(x-{na}))}\\$',(pa/nc)*infinity,'\\$ \\\\simplify{{(pa/nc)*infinity}}\\$'],\n  \n['\\$x\\\\rightarrow 0^+\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}sin(x))/({nc}cos(x)-{nc})}\\$','\\$\\\\simplify{({pa}cos(x))/({-nc}sin(x))}\\$',(-pa/nc)*infinity,'\\$ \\\\simplify{{(-pa/nc)*infinity}}\\$'],\n['\\$x\\\\rightarrow 0^-\\$','\\$\\\\frac{0}{0}\\$','\\$\\\\simplify{({pa}sin(x))/({nc}cos(x)-{nc})}\\$','\\$\\\\simplify{({pa}cos(x))/({-nc}sin(x))}\\$',(pa/nc)*infinity,'\\$ \\\\simplify{{(pa/nc)*infinity}}\\$']\n]", "templateType": "anything", "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>", "group": "Ungrouped variables"}, "log_tan": {"name": "log_tan", "definition": 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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>", "group": "Ungrouped variables"}, "nc": {"name": "nc", "definition": "nl[2]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "nL": {"name": "nL", "definition": "shuffle(-12..12 except 0)[0..5]", "templateType": "anything", "description": "", "group": "Ungrouped variables"}}, "ungrouped_variables": ["pL", "pa", "pb", "pc", "pd", "pe", "nL", "na", "nb", "nc", "nd", "ne", "alt1", "sin_poly", "cos_poly", "poly_sin", "poly_cos", "trig_trig", "log_sin", "log_cos", "log_tan", "sin_log", "cos_log", "tan_log", "choice1", "choice2"], "variablesTest": {"maxRuns": 100, "condition": ""}, "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/"}]}