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Question
Show that f(x) = |x| is continuous at x = 0.
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Solution
Given that f(x) = |x| = `{(x if x >= 2),(- x if x < 0):}`
`"L"["f"(x)]_(x=0) = lim_(x->0^-)`f(x)
[∵ x = 0 – h]
`= lim_(h->0^-) "f"(0 - "h")`
`= lim_(h->0^-) "f"(- "h")`
`= lim_(h->0^-) |- "h"|`
`= lim_(h->0^-) |"h"|`
`= lim_(h->0^-) "h" = 0`
`"R"["f"(x)]_(x=0^+) = lim_(x->0^+)`f(x)
`= lim_(h->0^+) "f"(0 - "h")`
`= lim_(h->0^+) "f"("h")`
`= lim_(h->0^+) |"h"|`
`= lim_(h->0) "h"`
= 0
[∵ |x| = x if x > 0]
Also f(0) = |0| = 0
`lim_(x->0^-) "f"(x) = lim_(x->0^+ "f"(x))` = f(0)
∴ f(x) is continuous at x = 0.
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