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Question
Evaluate the following limit :
`lim_(x -> 2)[(sqrt(1 + sqrt(2 + x)) - sqrt(3))/(x - 2)]`
Solution
`lim_(x -> 2)[(sqrt(1 + sqrt(2 + x)) - sqrt(3))/(x - 2)]`
`lim_(x -> 2) (sqrt(1 + sqrt(2 + x)) - sqrt(3))/(x - 2) xx (sqrt(1 + sqrt(2 + x)) + sqrt(3))/(sqrt(1 + sqrt(2 + x)) + sqrt(3))`
= `lim_(x -> 2) (1 + sqrt(2 + x) - 3)/((x - 2)(sqrt(1 + sqrt(2 + x)) + sqrt(3))`
= `lim_(x -> 2) (sqrt(2 + x) - 2)/((x - 2)(sqrt(1 + sqrt(2 + x)) + sqrt(3))) xx (sqrt(2 + x) + 2)/(sqrt(2 + x) + 2)`
= `lim_(x -> 2) ((2 + x) - 4)/((x - 2)(sqrt(1 + sqrt(2 + x)) + sqrt(3))(sqrt(2 + x) + 2))`
= `lim_(x -> 2) (x - 2)/((x - 2)(sqrt(1 + sqrt(2 + x)) + sqrt(3))(sqrt(2 + x) + 2))`
= `lim_(x -> 2) 1/((sqrt(1 + sqrt(2 + x)) + sqrt(3))(sqrt(2 + x) + 2)) ...[(because x -> 2"," x ≠ 2),(therefore x - 2 ≠ 0)]`
= `(lim_(x ->2) (1))/([lim_(x -> 2) (sqrt(1 + sqrt(2 + x)) + sqrt(3))] xx [lim_(x -> 2) (sqrt(2 + x) + 2)]`
= `1/((sqrt(1 + sqrt(4)) + sqrt(3))(sqrt(2 + 2) + 2)`
= `1/((2sqrt(3))(4))`
= `1/(8sqrt(3))`
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