limit as θ approaches 0 of θcos(3/θ)

{ "query": { "display": "$$\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{θ\\to 0}(θ\\cos(\\frac{3}{θ}))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)=0$$", "input": "\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the Squeeze Theorem:$${\\quad}0$$", "input": "\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)", "steps": [ { "type": "definition", "title": "Squeeze Theorem:", "text": "Let f, g and h be functions such that for all $$x\\in[a,\\:b]\\:$$(except possibly at the limit point c), <br/>$$f\\left(x\\right)\\le{h\\left(x\\right)}\\le{g\\left(x\\right)}$$<br/>Also suppose that, $$\\lim_{x\\to{c}}{f\\left(x\\right)}=\\lim_{x\\to{c}}{g\\left(x\\right)}=L$$<br/>Then for any $$a\\le{c}\\le{b},\\:\\lim_{x\\to{c}}{h\\left(x\\right)}=L$$", "secondary": [ "$$-1\\le\\:\\cos\\left(\\frac{3}{θ}\\right)\\le\\:1$$", "$$\\lim_{θ\\to\\:0}\\left(θ\\left(-1\\right)\\right)\\le\\:\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)\\le\\:\\lim_{θ\\to\\:0}\\left(θ1\\right)$$" ] }, { "type": "interim", "title": "$$\\lim_{θ\\to\\:0}\\left(θ\\left(-1\\right)\\right)=0$$", "input": "\\lim_{θ\\to\\:0}\\left(θ\\left(-1\\right)\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$θ=0$$", "result": "=0\\cdot\\:\\left(-1\\right)", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{θ\\to\\:0}\\left(θ\\cdot\\:1\\right)=0$$", "input": "\\lim_{θ\\to\\:0}\\left(θ\\cdot\\:1\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$θ=0$$", "result": "=0\\cdot\\:1", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "By the squeeze theorem: $$\\lim_{θ\\to\\:0}\\left(θ\\cos\\left(\\frac{3}{θ}\\right)\\right)=0$$", "result": "=0" } ], "meta": { "interimType": "Squeeze Theorem 0Eq", "practiceLink": "/practice/limits-practice?subTopic=Squeeze%20Theorem", "practiceTopic": "Limit Squeeze Theorem", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sQG5bCQ41/mmWHO9FMROz1WsXP+jJaeNhzdgCSYY8219PBH1SqfDFt1l4B7wN6SKFVYD8kpF1JNOZVCVjqr05i1EmtjMeblBEYh8D58AfqhCanqbe81B8fc1CEHBEBupEz/L0MoYg+CUn6oyL3EO7YrGddanbms4cigCpjwsLZ/8Gs3goJKBkDCRbiLpI7p8iZvpQ2FjMpW/ivxAYlJ5IjU=" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice?subTopic=Squeeze%20Theorem", "practiceTopic": "Limit Squeeze Theorem" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }