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Question
Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`
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Solution
Given `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`
`f(x) = {{:(|(x - 1|)/(x - 1), "for" x < 1 and x > 1),(0, "for" x = 1):}`
`f(x) = {{:((- (x - 1))/(x - 1), "for" x < 1),((x - 1)/(x - 1), "for" x > 1),(0, "for" x = 1):}`
`f(x) = {{:(-1, "for" x < 1),(1, "for" x > 1),(0, "for" x = 1):}`
`f(1^-) = lim_(x -> 1^-) f(x)`
= `lim_(x -> 1^-) (- 1)` = – 1 .......(1)
`f(1^+) = lim_(x -> 1^+) f(x)`
= `lim_(x -> 1^+) (1)` = 1 .......(2)
From equations (1) and (2) we get
f(1–) ≠ f(1+)
∴ The limit of f(x) does not exist.
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