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Question
Evaluate `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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