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Question
If f(x)= `{((x - |x|)/x if x ≠ 0),(2 if x = 0):}` then show that `lim_(x->1)`f(x) does not exist.
Solution
`"L"["f"(x)]_(x=0) = lim_(x->0^-) "f"(x) = lim_(h->0) "f"(0 - "h")`
`= lim_(h->0) "f"(-"h") = lim_(h->0) ((-"h") - |- "h"|)/(-"h")`
`= lim_(h->0) (- "h" - "h")/(- "h")`
`= lim_(h->0) (-2cancel("h"))/(-cancel(h))`
`= lim_(h->0) 2` = 2 ...[∵ |- h| = h] ...(1)
`"R"["f"(x)]_(x=0) = lim_(x->0) "f"(x) = lim_(x->0) "f"(0 + "h")`
`= lim_(h->0)` f(h)
`= lim_(h->0) ("h" - |"h"|)/"h"`
`= lim_(h->0) ("h - h")/"h"`
`= lim_(h->0) 0/"h"` = 0 ...(2)
From (1) and (2),
`"L"["f"(x)]_(x=0) ne "R"["f"(x)]_(x=0)`
`therefore lim_(x-> ∞) "f"(x)` does not exist.
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