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Question
Examine the continuity of `f(x) = {:((x^2 - 9)/(x - 3)",", "for" x ≠ 3),(=8",", "for" x = 3):}}` at x = 3.
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Solution
f(3) = 8 ...(1) (Given)
`lim_(x -> 3) f(x) = lim_(x -> 3) (x^2 - 9)/(x - 3)`
= `lim_(x -> 3) ((x - 3)(x + 3))/(x - 3)`
= `lim_(x -> 3) (x + 3)` ...[∵ x → 3, x ≠ 3, ∴ x – 3 ≠ 0]
= 3 + 3
= 6 ...(2)
From (1) and (2),
f(3) ≠ `lim_(x -> 3)` f(x)
∴ f is discontinuous at x = 3.
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