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Question
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
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Solution
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
= `lim_(x -> 0) ((1 + x)^(1/3) - (1 + x)^(1/2))/x`
Put 1 + x = y ⇒ x = y − 1
As x→0, y→1
∴ `lim_(x -> 0) ((1 + x)^(1/3) - (1 + x)^(1/2))/x`
= `lim_(y -> 1) (y^(1/3) - y^(1/2))/(y - 1)`
= `lim_(y -> 1) ((y^(1/3) - 1) - (y^(1/2) - 1))/(y - 1)`
= `lim_(y -> 1) ((y^(1/3) - 1)/(y - 1) - y^(1/2 - 1)/(y - 1))`
= `lim_(y -> 1) (y^(1/3) - 1^(1/3))/(y - 1) - lim_(y -> 1) (y^(1/2) - 1^(1/2))/(y - 1)`
= `1/3(1)^(-(2)/3) - 1/2(1)^(-(1)/2) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `1/3 - 1/2`
= `(2 - 3)/6`
= `-1/6`
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