limit as x approaches 1/3 of 3x^3+2x^2

{ "query": { "display": "$$\\lim_{x\\to\\:\\frac{1}{3}}\\left(3x^{3}+2x^{2}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to \\frac{1}{3}}(3x^{3}+2x^{2})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "\\frac{1}{3}", "decimal": "0.33333…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:\\frac{1}{3}}\\left(3x^{3}+2x^{2}\\right)=\\frac{1}{3}$$", "input": "\\lim_{x\\to\\:\\frac{1}{3}}\\left(3x^{3}+2x^{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=\\frac{1}{3}$$", "result": "=3\\left(\\frac{1}{3}\\right)^{3}+2\\left(\\frac{1}{3}\\right)^{2}", "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": "interim", "title": "Simplify $$3\\left(\\frac{1}{3}\\right)^{3}+2\\left(\\frac{1}{3}\\right)^{2}:{\\quad}\\frac{1}{3}$$", "input": "3\\left(\\frac{1}{3}\\right)^{3}+2\\left(\\frac{1}{3}\\right)^{2}", "result": "=\\frac{1}{3}", "steps": [ { "type": "interim", "title": "$$3\\left(\\frac{1}{3}\\right)^{3}=\\frac{1}{9}$$", "input": "3\\left(\\frac{1}{3}\\right)^{3}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{3}\\right)^{3}=\\frac{1}{3^{3}}$$", "input": "\\left(\\frac{1}{3}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{3}}{3^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{3}=1$$" ], "result": "=\\frac{1}{3^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oDCBTQiFZH4xk5D/XQxDlTEt0LhBzbgnHZdDPBpwndnMwViaLUXkeD+JukROhWdjV96xYdaKZxkiLMMT0B/wIf8//6/nV5O4fb8Xgwi7maoBHmcB8ZvUEK95ehaoXDKcACkvuZ0P56aQWLQ/MPQyg867uJicgMpLzW5Yu+HQ/jQ=" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{1}{3^{3}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{3^{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{3^{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1}{3^{2}}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\frac{1}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mMCTa8kvpqfbpq622pArQgIt3ueoOyqO+cp5lS+lCx9wkKGJWEPFPk38sdJMsyPIOeHHSRWuCicL1VPTZ6S+hDBKhlrUI4FrmISKcdB7IG2H4RUOKSrd4fH4aTg2Z+F68AY+bp6kM/DB23k5RXE7PiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$2\\left(\\frac{1}{3}\\right)^{2}=\\frac{2}{9}$$", "input": "2\\left(\\frac{1}{3}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{3}\\right)^{2}=\\frac{1}{3^{2}}$$", "input": "\\left(\\frac{1}{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{3^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{3^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oDCBTQiFZH4xk5D/XQxDlY5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdj0zsKgVxgZuct4fyCBKQvzf8//6/nV5O4fb8Xgwi7maoBHmcB8ZvUEK95ehaoXDKcC/NRIlHtcGFZ0+OZXJ3ZBFiVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{3^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{3^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{3^{2}}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\frac{2}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IMUZnkDqR6CTb04EC6GFFOiEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPI8tMXzFk06aeblP9klIhGQqFBoYgXYwU+YzV3MyHSB/RPS3PnluiX5oUnPOp71ZOfz6LjF+wuVgia9XQreJnG/yS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{1}{9}+\\frac{2}{9}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+2}{9}" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=\\frac{3}{9}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mMCTa8kvpqfbpq622pArQolYwPa1krSHMICogqjExBd+qfyA8gmlcAwV3PmoCpXuA585Wz2Y8ioMtXlAhbC3ecVwzg7RQQgvFH/N+iHBPccUmHg+iQxGlGxKWDb8ymJzHimBRYRqHSWeJkuUPhfTCwxANx1ekp+cu0L0ew7OXdMrebgNzNTU3jxycZMehYs/MJiACvrgq8D38WA2EwIYYw==" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }