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Question
Evaluate `lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)`
Solution
We have `lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)`
= `lim_(x -> 2) 1/(x - 2) (2(2x - 3))/(x(x - 1)(x - 2))`
= `lim_(x -> 2) (x(x - 1) - 2(2x - 3))/(x(x - 1)(x - 2))`
= `lim_(x -> 2) (x^2 - 5x + 6)/(x(x - 1)(x - 2))`
= `lim_(x -> 2) ((x - 2)(x - 3))/(x(x - 1)(x - 2))` .....[x – 2 ≠ 0]
= `lim_(x -> 2) (x - 3)/(x(x - 1)) = (-1)/2`
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