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Question
Evaluate the following limit :
`lim_(x -> 3)[(sqrt(2x + 3) - sqrt(4x - 3))/(x^2 - 9)]`
Solution
`lim_(x -> 3)[(sqrt(2x + 3) - sqrt(4x - 3))/(x^2 - 9)]`
`lim_(x -> 3)[(sqrt(2x + 3) - sqrt(4x - 3))/(x^2 - 9) xx (sqrt(2x + 3) + sqrt(4x - 3))/(sqrt(2x + 3) + sqrt(4x - 3))]`
= `lim_(x -> 3) [((2x + 3) - (4x - 3))/((x^2 - 9)(sqrt(2x + 3) + sqrt(4x - 3)))]`
= `lim_(x -> 3) [(-2x + 6)/((x^2 - 9)(sqrt(2x + 3) + sqrt(4x - 3)))]`
= `lim_(x -> 3)[(-2(x - 3))/((x + 3)(x - 3)(sqrt(2x + 3) + sqrt(4x - 3)))]`
= `lim_(x -> 3)[(-2)/((x + 3)(sqrt(2x + 3) + sqrt(4x - 3)))]` ...[∵ x → 3, ∴ x ≠ 3, ∴ x – 3 ≠ 0]
= `(-2)/((3 + 3)(sqrt(2(3) + 3) + sqrt(4(3) - 3))`
= `(-2)/(6(3 + 3))`
= `(-1)/(18)`
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