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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)[((x^(1/3) - 7^(1/3))(x^(1/3) + 7^(1/3)))/(x - 7)]`
= `lim_(x -> 7)[(x^(2/3) - 7^(2/3))/(x - 7)] ...[∵ (a – b)(a + b) = a^2 – b^2]`
= `2/3(7)^(2/3 - 1) ...[ therefore lim_(x -> a) (x^n - a^n)/(x - a) = na^(n - 1)]`
= `2/3(7)^(-1/3)`
= `2/3 × 1/7^(1/3)`
= `2/(3 (root(3)(7))`
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