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Question
Evaluate the following limits: `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))(sqrt(1 + x^2) + sqrt(1 + x)))/((sqrt(1 + x^3) - sqrt(1 + x))(sqrt(1 + x^3) + sqrt(1 + x))(sqrt(1 + x^2) + 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(x - 1)(sqrt(1 + x^3) + sqrt(1 + x)))/(x(x^2 - 1)(sqrt(1 + x^2) + sqrt(1 + x))`
= `lim_(x -> 0) (x(x - 1)(sqrt(1 + x^3) + sqrt(1 + x)))/(x(x - 1)(x + 1)(sqrt(1 + x^2) + sqrt(1 + x)))`
= `lim_(x -> 0) (sqrt(1 + x^3) + sqrt(1 + x))/((x + 1)(sqrt(1 + x^2) + sqrt(1 + x))` ...[∵ x → 0, ∴ x ≠ 0, ∴ x – 1 ≠ 0]
= `(sqrt(1 + 0^3) + sqrt(1 + 0))/((0 + 1)(sqrt(1 + 0^2) + sqrt(1 + 0))`
= `(1 + 1)/(1(1 + 1)`
= 1
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