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Question
Evaluate the following limit :
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
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Solution
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
= `lim_(x -> 7) ((root(3)(x))^2 - (root(3)(7))^2)/(x - 7)`
= `lim_(x -> 7) (x^(2/3) - 7^(2/3))/(x - 7)`
= `2/3(7)^(-1/3) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `2/(3(root(3)(7))`.
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