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Question
Evaluate: `lim_(x -> 0) (sqrt(1 + x^3) - sqrt(1 - x^3))/x^2`
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Solution
Given that `lim_(x -> 0) (sqrt(1 + x^3) - sqrt(1 - x^3))/x^2`
= `lim_(x -> 0) ([sqrt(1 + x^3) - sqrt(1 - x^3)][sqrt(1 + x^3) + sqrt(1 - x^3)])/(x^2[sqrt(1 + x^3) + sqrt(1 - x^2)])`
= `lim_(x -> 0) ((1 + x^3) - (1 - x^3))/(x^2[sqrt(1 + x^3) + sqrt(1 - x^3)])`
= `lim_(x -> 0) (1 + x^3 - 1 + x^3)/(x^2[sqrt(1 + x^3) + sqrt(1 - x^3)])`
= `lim_(x -> 0) (2x^3)/(x^2[sqrt(1 + x^3) + sqrt(1 - x^3)])`
= `lim_(x -> 0) (2x)/(sqrt(1 + x^3) + sqrt(1 - x^3)`
= 0
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