limit as x approaches 0 of-arcsin(x)

{ "query": { "display": "$$\\lim_{x\\to\\:0}\\left(-\\arcsin\\left(x\\right)\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to 0}(-\\arcsin(x))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:0}\\left(-\\arcsin\\left(x\\right)\\right)=0$$", "input": "\\lim_{x\\to\\:0}\\left(-\\arcsin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=-\\arcsin\\left(0\\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": "interim", "title": "Simplify $$-\\arcsin\\left(0\\right):{\\quad}0$$", "input": "-\\arcsin\\left(0\\right)", "result": "=0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\arcsin\\left(0\\right)=0$$", "secondary": [ "$$\\begin{array}{|c|c|c|}\\hline x&\\arcsin(x)&\\arcsin(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{1}{2}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline \\frac{\\sqrt{2}}{2}&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{2}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline 1&\\frac{\\pi}{2}&90^{\\circ}\\\\\\hline \\end{array}$$" ], "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f0aEXGdi9dnJbps05BXTQ1XTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZW72wZm7kDUxdE6YSmfEbr2lT0KpnuVm9gQ6pQiT6Pi/NBGop4RgYHNuC0ZBqQQZK8" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }