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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
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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