f(x)=2arctan(x)

{ "query": { "display": "$$f\\left(x\\right)=2\\arctan\\left(x\\right)$$", "symbolab_question": "FUNCTION#f(x)=2\\arctan(x)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{Domain}: -\\infty <x<\\infty <br/>\\mathrm{Range}: -π<f(x)<π<br/>\\mathrm{X\\:Intercepts}: (0,0),\\mathrm{Y\\:Intercepts}: (0,0)<br/>\\mathrm{Asymptotes}: \\mathrm{Horizontal}\\:y=π,y=-π", "interval": "\\mathrm{Domain}: (-\\infty ,\\infty )<br/>\\mathrm{Range}: (-π,π)" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Domain of $$2\\arctan\\left(x\\right)\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$2\\arctan\\left(x\\right):{\\quad}-π<f\\left(x\\right)<π$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "The range of the basic $$\\arctan\\:$$function is $$-\\frac{π}{2}<\\arctan\\left(x\\right)<\\frac{π}{2}$$", "result": "-\\frac{π}{2}<\\arctan\\left(x\\right)<\\frac{π}{2}" }, { "type": "step", "primary": "Multiply the edges of the range by: $$2$$", "result": "-π<2\\arctan\\left(x\\right)<π" }, { "type": "step", "primary": "Therefore the range is", "result": "-π<f\\left(x\\right)<π" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Simple Trig Top 1Eq" } }, { "type": "interim", "title": "Axis interception points of $$2\\arctan\\left(x\\right):\\quad\\:$$X Intercepts$$:\\:\\left(0,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$2\\arctan\\left(x\\right):{\\quad}\\left(0,\\:0\\right)$$", "input": "2\\arctan\\left(x\\right)", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "$$2\\arctan\\left(x\\right)=0{\\quad:\\quad}x=0$$", "input": "2\\arctan\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\arctan\\left(x\\right)=0", "result": "\\arctan\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\arctan\\left(x\\right)}{2}=\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\arctan\\left(x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+t/VmUDe0iC/MSgiZR/7A/fJNP+zB1NTqjBnG4FP0kFOdta0v9i0ue5RTdWUc9s5LI68O3OdsD4eSYemA2sDCGp7t9VOlxVwXev7QS8K3g+bUbIlJ+t81LQlTGDDIvaLsoh1N1bMWdCb5YM9Q4f9ltMyKXqjig1oe8nVG8YCoXV694Ph869lyDoQUdss535rb5ffsy7dLc7Kj7Kf73aSsyHz9y2wQ6GMyxViP0+v27A3YCGz30lX7Q0UBNnN4RNTflQkXR+Ml7/yW3CuFrYQtICcIpnsl65bTpN7k6v/0tv1oT/Ge5uSbL7/vI8VF1smf+66RtCVxcLcJy61gPGXq6XCQEd42wbq97zFf9SJ0v9PGK5kkXo6LlenBYTcE1DA1NygGOJAx4pTOq+OWjGJoUU+hbyKYvkiHPPqN4EfLdWLXbop+UgZc+PRPrpJXemYAidZUNRQMFY0epg1rbkx+Q4A5ZNYv+MgLRJ/z3BVeMCjeh7+jKEzLb7VNCEMF3Z/CJnvlR8hNcW0Sh62aThkwHs4Yjn2RnGQgX/dNKAnX/Qkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\arctan\\left(x\\right)=0", "result": "x=0", "steps": [ { "type": "step", "primary": "$$\\arctan\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\tan\\left(a\\right)$$", "result": "x=\\tan\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan\\left(0\\right)=0$$", "input": "\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "input": "\\tan\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "result": "x=0" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7NUgz5f0jydS6eikA8Jpty2vjRQaCcgxOEEl56sUtiypTCJtI9vvSu8YNGFp1F5HQKG9V3uxwLjAeN24uzvbXNYEK3rgSSRFh70SoYtmyxwabCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$2\\arctan\\left(x\\right):{\\quad}\\left(0,\\:0\\right)$$", "input": "2\\arctan\\left(x\\right)", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=2\\arctan\\left(0\\right):{\\quad}0$$", "input": "2\\arctan\\left(0\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arctan\\left(0\\right)=0$$", "input": "\\arctan\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arctan(x)&\\arctan(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{3}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline 1&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\sqrt{3}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfkOM8wwjRYm+rJJdfSn/Mzv6oiBBnyfG5CYjXjL/RFPpaWVX6xzKg+mqD2grc5O15roTIhNaKeFmvNue7ok79ZL7nYcMBEbbEHcQso51o3o2CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(0,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Asymptotes of $$2\\arctan\\left(x\\right):\\quad\\:$$Horizontal$$:\\:y=π,\\:y=-π$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$2\\arctan\\left(x\\right):{\\quad}$$None", "input": "2\\arctan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Go over every undefined point $$x=a$$ and check if at least one of the following statements is true:<br/>$${\\quad}\\lim_{x\\to{a^{-}}}f\\left(x\\right)=\\pm\\infty$$<br/>$${\\quad}\\lim_{x\\to{a^{+}}}f\\left(x\\right)=\\pm\\infty$$" }, { "type": "step", "primary": "The function $$2\\arctan\\left(x\\right)\\:$$has no undefined points" }, { "type": "step", "result": "\\mathrm{No\\:vertical\\:asymptotes}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5deCxnl6SZHpGlVu97Meq5ZsPPzH7B/TPKnh5KdC00QxGY3Nayl+T2frFdHjgW2XADmzfXRBU6TSklKt4GXvQTtV/rYJoVElfTCpO++C6qax88xG3urWPe2/B1+jqORv8dg==" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$2\\arctan\\left(x\\right):{\\quad}y=π,\\:y=-π$$", "input": "2\\arctan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Check if at $$x\\to\\pm\\infty$$ the function $$y=2\\arctan\\left(x\\right)$$ behaves as a line, $$y=b$$" }, { "type": "interim", "title": "Find an asymptote for $$x\\to-\\infty\\::{\\quad}y=-π$$", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to-\\infty\\:}{f\\left(x\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{x\\to-\\infty\\:}{f\\left(x\\right)}=\\lim_{x\\to-\\infty\\:}{2\\arctan\\left(x\\right)}=-π$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(2\\arctan\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=-\\frac{π}{2}$$", "result": "=2\\left(-\\frac{π}{2}\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(-\\frac{π}{2}\\right):{\\quad}-π$$", "input": "2\\left(-\\frac{π}{2}\\right)", "result": "=-π", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-2\\cdot\\:\\frac{π}{2}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-π" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sWrPcXp/WY1rFdR2GNUxYa7tFg3p9ywqfD69DXG2UbWjkVi15I8rBefLi4Iyt2wrdypIzQk6zOl4kTxnC1+aw+5AIz++qluupTlLFEcE9J1mXUa9LileZ7p4vtNNpbqIRFqWiAAeC6p8tZOVoL5iMQ==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$x\\to\\infty\\::{\\quad}y=π$$", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to\\infty\\:}{f\\left(x\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\infty\\:}{f\\left(x\\right)}=\\lim_{x\\to\\infty\\:}{2\\arctan\\left(x\\right)}=π$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(2\\arctan\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=\\frac{π}{2}$$", "result": "=2\\cdot\\:\\frac{π}{2}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{2}:{\\quad}π$$", "input": "2\\cdot\\:\\frac{π}{2}", "result": "=π", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=π" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvijDjqYwMGUrS2638vECqwzQJQJZuTAY5js+oqjdT8ksl58bIKwWzS3dUPXOsTsmj0h429vuTSxWa7B/X3D1oP01aFhYiqspO750q8e++FI568XMVGoNNJylSL2Q0+ebauCS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=π,\\:y=-π" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0Pihok+TXzE4gZrESDQtn05OY+5/fsIljUUYxvHKKNy3xWn6TeQKHeh69S6dnv9vSoUoFPGjsFp2QNKCOLceuhhwMilkpBEi1foGyJlQwGqR3WSdy4KVQCcaWZPjRUgsgrC6Sw==" } }, { "type": "interim", "title": "Slant Asymptotes of $$2\\arctan\\left(x\\right):{\\quad}$$None", "input": "2\\arctan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Check if at $$x\\to\\pm\\infty$$ the function $$y=2\\arctan\\left(x\\right)$$ behaves as a line, $$y=mx+b\\:$$ where $$m\\neq0$$" }, { "type": "interim", "title": "Find an asymptote for $$x\\to-\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to-\\infty\\:}{\\frac{f\\left(x\\right)}{x}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{x\\to-\\infty\\:}{\\frac{f\\left(x\\right)}{x}}=\\lim_{x\\to-\\infty\\:}{\\frac{2\\arctan\\left(x\\right)}{x}}=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{2\\arctan\\left(x\\right)}{x}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{\\arctan\\left(x\\right)}{x}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=2\\cdot\\:\\frac{\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)}{\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=-\\frac{π}{2}$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=-\\frac{π}{2}$$", "result": "=-\\frac{π}{2}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)=-\\infty\\:$$", "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{-\\frac{π}{2}}{-\\infty\\:}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{-\\frac{π}{2}}{-\\infty\\:}:{\\quad}0$$", "input": "2\\cdot\\:\\frac{-\\frac{π}{2}}{-\\infty\\:}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\frac{c}{-\\infty}=0$$", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvikM+00K8YX9mPcvxq5OWXP2q1oFQqSQZRaVx6gxydWVi3XeO2tIUPH5Q2xrCOU6NXU2wDsU/MDk5bMOYBn/h3a/WwPs1+Gw97t4MeuaNjSYTTtF3u+uPIVjW4FW9gEh6txckOUHpQIiFqG93j0abwmtJHFoGi6y31eG7eErdf9g3t6U35iUjp6KMcdrtShMOQQ==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is zero, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$x\\to\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to\\infty\\:}{\\frac{f\\left(x\\right)}{x}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\infty\\:}{\\frac{f\\left(x\\right)}{x}}=\\lim_{x\\to\\infty\\:}{\\frac{2\\arctan\\left(x\\right)}{x}}=0$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{2\\arctan\\left(x\\right)}{x}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2\\cdot\\:\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\arctan\\left(x\\right)}{x}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=2\\cdot\\:\\frac{\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)}{\\lim_{x\\to\\:\\infty\\:}\\left(x\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=\\frac{π}{2}$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(\\arctan\\left(x\\right)\\right)=\\frac{π}{2}$$", "result": "=\\frac{π}{2}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(x\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(x\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{\\frac{π}{2}}{\\infty\\:}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{\\frac{π}{2}}{\\infty\\:}:{\\quad}0$$", "input": "2\\cdot\\:\\frac{\\frac{π}{2}}{\\infty\\:}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\frac{c}{\\infty}=0$$", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviuuwsq2wQ+rqB8Tj3xWbkGqNpMEUEcpr+5iWc+duEvRIA585Wz2Y8ioMtXlAhbC3ebgNL2ycNF0YUgQmiI0KpBZN5Aod6Hr1Lp2e/29KhSgU4nw7BGf6PVvnUoBojhEDgFH2gANAFAwvB5mysKrmEuGtUk3DVL+jhRTT8Q/Yh1lHialcV/dI5TH4fXyp+ncwuA==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is zero, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KJrxvQn/y/6CVqHGYJldXwjne/KkjTlXnA/R2E0AkYizHgtCjLJHuadi4Y6DgWLehTt1PvLCDyaUc9lUx4FFrfOQORIgQR9ULVYKTDBn0EIj" } }, { "type": "step", "result": "\\mathrm{Horizontal}:\\:y=π,\\:y=-π" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Function Asymptotes Top 2Eq" } } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\arctan(x)" }, "showViewLarger": true } } }