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Question
Find `lim_(x -> 5) f(x)`, where f(x) = |x| - 5
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Solution
`f(x) = |x| - 5`
`lim_(x → 5^-) f(x) = lim_(x → 5^-) [|x| - 5]`
= `lim_(h → 0) [|5 - h| - 5]`
= `lim_(h → 0) [5 - h - 5] = lim_(h → 0) (-h) = 0`
= `lim_(x → 5^+) f(x) = lim_(x → 5^+) [|x| - 5] = lim_(h → 0) [|5 + h| - 5]`
= `lim_(h → 0) [5 + h - 5] = lim_(h → 0) h = 0`
∴ `lim_(x → 5^-) f(x) = lim_(x → 5^+) f(x)`
∴ `lim_(x → 5) f(x) = 0`
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