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Question
Evaluate the following limits: `lim_(x -> 2)[(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`
Solution
`lim_(x -> 2)[(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`
= `lim_(x -> 2) [(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2)) xx (sqrt(2 + x) + sqrt(6 - x))/(sqrt(2 + x) + sqrt(6 - x)) xx (sqrt(x) + sqrt(2))/(sqrt(x) + sqrt(2))] ...[("By taking conjuagates"),("of both, the numerator"),("as well as the"),("Denomerator")]`
= `lim_(x -> 2) [((2 + x) - (6 - x))/((x - 2)) xx (sqrt(x) + sqrt(2))/(sqrt(2 + x) + sqrt(6 - x))]`
= `lim_(x -> 2) [(-4 + 2x)/(x - 2) xx (sqrt(x) + sqrt(2))/(sqrt(2 + x) + sqrt(6 - x))]`
= `lim_(x -> 2) [(2(x - 2))/(x - 2) xx (sqrt(x) + sqrt(2))/(sqrt(2 + x) + sqrt(6 - x))]`
= `lim_(x -> 2) [(2(sqrt(x) + sqrt(2)))/(sqrt(2 + x) + sqrt(6 - x))] ...[(because x -> 2";" x ≠ 2),(therefore x - 2 ≠ 0)]`
= `(2(sqrt(2) + sqrt(2)))/(sqrt(2 + 2) + sqrt(6 - 2)`
= `(2(2sqrt(2)))/(2 + 2)`
= `(4sqrt(2))/4`
= `sqrt(2)`
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