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Question
Evaluate the following Limits: `lim_(x -> 3)[(x - 3)/(sqrt(x - 2) - sqrt(4 - x))]`
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Solution
`lim_(x -> 3)[(x - 3)/(sqrt(x - 2) - sqrt(4 - x))]`
= `lim_(x -> 3)[(x - 3)/(sqrt(x - 2) - sqrt(4 - x)) xx(sqrt(x - 2) + sqrt(4 - x))/(sqrt(x - 2) + sqrt(4 - x))]`
= `lim_(x ->3) ((x - 3)(sqrt(x - 2) + sqrt(4 - x)))/((x - 2) - (4 - x)`
= `lim_(x -> 3) ((x - 3)(sqrt(x - 2) + sqrt(4 - x)))/(2x - 6)`
= `lim_(x -> 3) ((x - 3)(sqrt(x - 2) + sqrt(4 - x)))/(2(x - 3)`
= `lim_(x -> 3) (sqrt(x - 2) + sqrt(4 - x))/2 ...[("As" x -> 3"," x ≠ 3),(therefore x - 3 ≠ 0)]`
= `1/2 lim_(x -> 3)(sqrt(x - 2) + sqrt(4 - x))`
= `1/2(sqrt(3 - 2) + sqrt(4 - 3))`
= `1/2(1 + 1)`
= 1
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