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Question
Evaluate the following limits: `lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
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
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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