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Question
Examine the following function for continuity:
f(x) = `1/(x - 5)`, x ≠ 5
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Solution
Let a be a real number, then,
`lim_(x->a^+)` f(x) = `lim_(h->0) 1/(a + h - 5) = 1/(a - 5)`
`lim_(x->a^-)` f(x) = `lim_(h->0) 1/(a - h - 5) = 1/(a-5)`
f(a) = `1/(a-5)`
`∵ lim_(x->a^+)` f(x) = `lim_(x->a^-)` f(x) = f(a)
Hence, the given function f(x) = `1/(x - 5)` is continuous at all points except at x = 5.
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