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Question
Find `lim_(x -> 0)` f(x), where `f(x) = {(x/|x|, x != 0),(0, x = 0):}`
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Solution
If x < 0, |x| = −x
∴ `lim_(x → 0^-) f(x) = lim_(x → 0^-) (x/|x|) = lim_(x → 0^-)(-x)/x = -1`
And if x > 0, |x| = x
∴ `lim_(x → 0^+) f(x) = lim_(x → 0^+) (x/|x|) = lim_(x → 0^+) x/x = 1`
∴ `lim_(x → 0^-) f(x) ≠ lim_(x → 0^+) f(x)`
Hence, the equation does not exist at x = 0.
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