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Question
`lim_(x->∞) [x{sqrt(x^2+1) - sqrt(x^2-1)}]`
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Solution
`lim_(x->∞) [x{sqrt(x^2+1) - sqrt(x^2-1)}]`
Step 1: Multiply and divide by the conjugate
`x(sqrt(x^2+1)-sqrt(x^2-1)) xx (sqrt(x^2+1)+sqrt(x^2-1))/(sqrt(x^2+1)+sqrt(x^2-1))`
`x xx ((x^2+1) - (x^2-1))/(sqrt(x^2+1) + sqrt(x^2-1))`
(x2 + 1) − (x2 − 1) = 2
`= (2x)/(sqrt(x^2+1)+sqrt(x^2-1)`
Step 2: Factor out xxx from the denominator
`= (2x)/(xsqrt(1+1/x^2) + xsqrt(1-1/x^2))`
`= (2x)/(xsqrt(1+1/x^2) + sqrt(1-1/x^2))`
`= (2)/(sqrt(1+1/x^2) + sqrt(1-1/x^2))`
Step 3: Take the limit as x → ∞
`1/x^2 -> 0`
`sqrt(1+1/x^2) ->1, sqrt(1-1/x^2) ->1`
`2/(1+1)`
`2/2`
= 1
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