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प्रश्न
Evaluate the following limits: `lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))]`
उत्तर
`lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))]`
= `lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2)) xx (sqrt(x + 2) + sqrt(3x - 2))/(sqrt(x + 2) + sqrt(3x - 2))]`
= `lim_(x -> 2) ((x^3 - 8)(sqrt(x + 2) + sqrt(3x - 2)))/((x + 2) - (3x - 2)`
= `lim_(x -> 2) ((x^3 - 8)(sqrt(x + 2) + sqrt(3x - 2)))/(-2x + 4)`
= `lim_(x -> 2) ((x - 2)(x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2(x - 2))`
= `lim_(x -> 2) ((x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2) ...[("As" x -> 2"," x ≠ 2),(therefore x - 2 ≠)]`
= `-1/2 lim_(x -> 2) (x^2 + 2x + 4) xx lim_(x -> 2) (sqrt(x + 2) + sqrt(3x - 2))`
= `-1/2 xx [(2)^2 + 2(2) + 4] xx (sqrt(2 + 2) + sqrt(3(2) - 2))`
= `-1/2 xx 12 xx (2 + 2)`
= – 6 x 4
= – 24
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