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Question
Examine whether the function is continuous at the points indicated against them:
f(x) = `(x^2 + 18x - 19)/(x - 1)` for x ≠ 1
= 20 for x = 1, at x = 1
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Solution
`lim_(x→1) "f"(x) = lim_(x→1) (x^2 + 18x - 19)/(x - 1)`
= `lim_(x→1) (x^2 + 19x - x - 19)/(x - 1)`
= `lim_(x→1) (x(x + 19) - 1(x + 19))/(x - 1)`
= `lim_(x→1) ((x - 1)(x + 19))/((x - 1))`
= `lim_(x→1) (x + 19)` ....[∵ x → 1, ∴ x ≠ 1, ∴ x - 1 ≠ 0]
= 1 + 19 = 20
Also, f(1) = 20
∴ `lim_(x→1) "f"(x) = "f"(1)`
∴ f(x) is continuous at x = 1
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