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Question
Evaluate the following limits: `lim_(x -> 2)[(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`
Solution
`lim_(x -> 2)[(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`
= `lim_(x -> 2) [(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2)) xx(sqrt(x + 2) + sqrt(3x - 2))/(sqrt(x + 2) + sqrt(3x - 2))]`
= `lim_(x -> 2) ((x^2 - 4)(sqrt(x + 2) + sqrt(3x - 2)))/((x + 2) - (3x - 2)`
= `lim_(x -> 2) ((x^2 - 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2x + 4)`
= `lim_(x -> 2) ((x + 2)(x - 2)(sqrt(x + 2) + sqrt(3x - 2)))/(-2(x - 2)`
= `lim_(x -> 2) ((x + 2)(sqrt(x + 2) + sqrt(3x - 2)))/(-2) ...[("As" x -> 2"," x ≠ 2),(therefore x - 2 ≠0)]`
= `((2 + 2)(sqrt(2 + 2) + sqrt(3(2) - 2)))/(-2)`
= `(4(2 + 2))/(-2)`
= `16/(-2)`
= – 8
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