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Question
I am taking 11th grade Math, but I am trying to do some learning on my own and work ahead into some calculus stuff (super-nerd). I am having trouble with this problem, dealing with find limits:

Find the limit of |x|/x (absolute value of x, divided by x) when x->0 (I believe x->0 means when x approaches 0?)

Answer
The expression

lim x->0 of |x|/x

actually does not have a limit in the usual sense of approaching a single value when x gets closer and closer to zero from both the positive, x>0, and negative, x<0, sides. The problem is that the expression must approach a single value in a neighborhood around x = 0. A neighborhood is normally defined as values of x on both sides of x = 0. Since

|x|/x = x/x = 1 for x >0

and

|x|/x = -|x|/|x| = -1 for x < 0

no matter how close x gets to 0, the expression is discontinuous at x = 0 and does not have a limit.

However, it is OK to take right hand and left hand limits. A right hand limit is where x > 0+ from the positive side and the left hand limit is x > 0- from the negative side. In these cases, |x|/x -> 1 and -1 respectively.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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