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Question
Test the continuity of the following function at the point or interval indicated against them:
f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",", "for" 0 < x < 3"," x ≠ 2),(= 12",", "for" x = 2),(= (2 - 2x - x^2)/(x - 4)",", "for" 3 ≤ x < 4):}}` at x = 2
Solution
f(2) = 12 ...(Given) ...(1)
`lim_(x -> 2) "f"(x) = lim_(x -> 2) (x^2 + 8x - 20)/(2x^2 - 9x + 10)`
= `lim_(x -> 2) ((x - 2)(x + 10))/((x - 2)(2x - 5))`
= `lim_(x -> 2) (x + 10)/(2x - 5)` ...`[(∵ x → 2 x ≠ 2),(∴ x → 2 ≠ 0)]`
= `(lim_(x -> 2) (x + 10))/(lim_(x -> 2) (2x - 5))`
= `(2 + 10)/(4 - 5)`
= – 12 ...(2)
From (1) and (2),
`lim_(x -> 2) "f"(x) ≠ f(2)`
∴ f is discontinuous at x = 2.
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