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Question
Evaluate the following :
`lim_(x -> ∞) [sqrt(x)(sqrt(x + 1) - sqrt(x))]`
Solution
`lim_(x -> ∞) [sqrt(x)(sqrt(x + 1) - sqrt(x))]`
= `lim_(x -> ∞) sqrt(x) (sqrt(x + 1) - sqrt(x)) xx ((sqrt(x + 1) + sqrt(x)))/(sqrt(x + 1) + sqrt(x))`
= `lim_(x -> ∞) (sqrt(x)(x + 1 - x))/(sqrt(x + 1) + sqrt(x))`
= `lim_(x -> ∞) (sqrt(x))/(sqrt(x + 1) + sqrt(x))`
= `lim_(x -> ∞) 1/((sqrt(x + 1))/(sqrt(x)) + 1) ...[("Divide numerator and"),("denominator by" sqrt(x))]`
= `lim_(x -> ∞) 1/(sqrt((x + 1)/x) + 1)`
= `1/(lim_(x -> ∞)(sqrt(1 + 1/x ) + 1))`
= `1/(sqrt(1 + 0) + 1) ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= `1/(1 + 1)`
= `1/2`
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