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Question
Evaluate the following :
`lim_(x -> ∞) [sqrt(x^4 + 4x^2) - x^2]`
Solution
Let L = `lim_(x -> ∞) [sqrt(x^4 + 4x^2) - x^2]`
= `lim_(x -> ∞) [((sqrt(x^4 + 4x^2) - x^2) xx (sqrt(x^4 + 4x^2) + x^2))/(sqrt(x^4 + 4x^2) + x^2)]`
= `lim_(x -> ∞) ((x^4 + 4x^2 - x^4))/(sqrt(x^4 + 4x^2) + x^2)`
= `lim_(x -> ∞) (4x^2)/(sqrt(x^4 (1 + 4/x^2)) + x^2`
= `lim_(x -> ∞) (4x^2)/(x^2sqrt(1 + 4/x^2) + x^2`
Dividing numerator and denominator by x2, we get,
L = `lim_(x -> ∞) 4/(sqrt(1 + 4/x^2) + 1)`
= `(lim_(x -> ∞) 4)/(lim_(x -> ∞) sqrt(1 + 4 xx 1/x^2 + lim_(x -> ∞) 1)`
= `4/(sqrt(1 + 4 xx 0) + 1) ...[because lim_(x -> ∞) 1/x^"n" = 0 "if n" > 0]`
= 2
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