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Question
Evaluate the Following limit :
`lim_(x -> 0) [(sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 - x))]`
Solution
`lim_(x -> 0) (sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 - x))`
= `lim_(x -> 0) (sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 + x)) xx(sqrt(1 + x^2) + sqrt(1 + x))/(sqrt(1 + x^2) + sqrt(1 + x)) xx (sqrt(1 + x^3) + sqrt(1 + x))/(sqrt(1 + x^3) + sqrt(1 + x))`
= `lim_(x -> 0) ((1 + x^2 - 1 - x)(sqrt(1 + x^3) + sqrt(1 + x)))/((1 + x^3 - 1 - x)(sqrt(1 + x^2) + sqrt(1 + x))`
= `lim_(x -> 0) ((x^2 - x)(sqrt(1 + x^3) + sqrt(1 + x)))/((x^3 - x)(sqrt(1 + x^2) + sqrt(1 + x))`
= `lim_(x -> 0) (x(x - 1)(sqrt(1 + x^3) + sqrt(1 + x)))/(x(x^2 - 1)(sqrt(1 + x^2) + sqrt(1 + x))`
= `lim_(x -> 0) ((x - 1)(sqrt(1 + x^3) + sqrt(1 + x)))/((x^2 - 1)(sqrt(1 + x^2) + sqrt(1 + x))` ...[∵ x → 0 ∴ x ≠ 0]
= `(lim_(x -> 0) (x - 1) xx lim_(x -> 0) (sqrt(1 + x^3) + sqrt(1 + x)))/(lim_(x -> 0) (x^2 - 1) xx lim_(x -> 0) (sqrt(1 + x^2) + sqrt(1 + x))`
= `((0 - 1)(1 + 1))/((0 - 1)(1 + 1))`
= 1
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