limit as x approaches 0-of 1/x-1/(|x|)

{ "query": { "display": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}-\\frac{1}{\\left|x\\right|}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to 0-}(\\frac{1}{x}-\\frac{1}{\\left|x\\right|})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "-\\infty ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}-\\frac{1}{\\left|x\\right|}\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}-\\frac{1}{\\left|x\\right|}\\right)", "steps": [ { "type": "step", "primary": "$$x$$ is negative when $$x\\to\\:0-$$. Therefore $$\\left|x\\right|=-x$$", "result": "=\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}-\\frac{1}{-x}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}\\right)-\\lim_{x\\to\\:0-}\\left(\\frac{1}{-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": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the left$$,\\:x<0\\quad\\Rightarrow\\quad\\:x<0$$", "secondary": [ "The denominator is a negative quantity approaching 0 from the left" ], "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{-x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{1}{-x}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the left$$,\\:x<0\\quad\\Rightarrow\\quad\\:-x>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=-\\infty\\:-\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$-\\infty-\\infty=-\\infty$$", "result": "=-\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }