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Question
Evaluate: `lim_(x -> 0) ((x + 2)^(1/3) - 2^(1/3))/x`
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Solution
Given that `lim_(x -> 0) ((x + 2)^(1/3) - 2^(1/3))/x`
Put x + 2 = y
⇒ x = y – 2
= `lim_(y - 2 -> 0) (y^(1/3) - 2^(1/3))/(y - 2)`
= `lim_(y -> 2) (y^(1/3) - 2^(1/3))/(y - 2)`
= `1/3 * (2)^(1/3 - 1)`
= `1/3 * 2^((-2)/3)` ......`["Using" lim_(x -> a) (x^n - a^n)/(x - a) = n * a^(n - 1)]`
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