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Question
Evaluate the following limit :
`lim_(x -> 1) [(x + 2)/(x^2 - 5x + 4) + (x - 4)/(3(x^2 - 3x + 2))]`
Solution
`lim_(x -> 1) [(x + 2)/(x^2 - 5x + 4) + (x - 4)/(3(x^2 - 3x + 2))]`
= `lim_(x -> 1) [(x + 2)/((x - 1)(x - 4)) + (x - 4)/(3(x - 1)(x - 2))]`
= `lim_(x -> 1) [(3(x + 2)(x - 2) + (x - 4)^2)/(3(x - 1)(x - 2)(x - 4))]`
= `lim_(x -> 1) (3x^2 - 12 + x^2 - 8x + 16)/(3(x - 1)(x - 2)(x - 4)`
= `lim_(x -> 1) (4x^2 - 8x + 4)/(3(x - 1)(x - 2)(x - 4)`
= `lim_(x -> 1) (4(x - 1)^2)/(3(x - 1)(x - 2)(x - 4)`
= `lim_(x -> 1)(4(x - 1))/(3(x^2 - 6x + 8)) ...[(because x -> 1"," x ≠ 1),(therefore x - 1 ≠ 0)]`
= `(lim_(x -> 1) [4(x - 1)]]/(lim_(x -> 1)[3(x^2 - 6x + 8)]`
= `(4(1 - 1))/(3(1 - 6 + 8))`
= 0
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