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प्रश्न
Evaluate the following limit :
`lim_(x -> 1) [(x - 2)/(x^2 - x) - 1/(x^3 - 3x^2 + 2x)]`
उत्तर
`lim_(x -> 1) [(x - 2)/(x^2 - x) - 1/(x^3 - 3x^2 + 2x)]`
= `lim_(x -> 1) [(x - 2)/(x(x - 1)) - 1/(x(x^2 - 3x + 2))]`
= `lim_(x -> 1) [(x - 2)/(x(x - 1)) - 1/(x(x - 1)(x - 2))]`
= `lim_(x -> 1) ((x - 2)^2 - 1^2)/(x(x - 1)(x - 2))`
= `lim_(x -> 1) ((x - 2 + 1)(x - 2 - 1))/(x(x - 1)(x - 2))`
= `lim_(x -> 1) ((x - 1)(x - 3))/(x(x - 1)(x - 2))`
= `lim_(x -> 1) (x - 3)/(x^2 - 2x) ...[(because x -> 1"," x ≠ 1),(therefore x - 1 ≠ 0)]`
= `(lim_(x -> 1) (x - 3))/(lim_(x -> 1)(x^2 - 2x))`
= `(1 - 3)/(1^2 - 2(1))`
= 2.
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