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Question
Evaluate the following limits:
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x`
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Solution
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x = lim_(x -> oo) ((x^2 - 4x + 2 + 2x - 1)/(x^2 - 4x + 2))^x`
= `lim_(x -> oo) [(x^2 - 4x - 2)/(x^2 - 4x + 2) +(2x - 1)/(x^2 - 4x + 2)]^x`
= `lim_(x -> oo) [1 + (2x - 1)/(x^2 - 4x + 2)]^x`
= `lim_(x - oo) [1 + 1/((x^2 -4x + 2)/(2x - 1))]^(((x^2 - 4x + 2)/(2x - 1) xx ((2x - 1)x)/(x^2 - 4x + 2))`
= `lim_(x -> oo) [(1 + (2x - 1)/(x^2 - 4x + 2))^((x^2 - 4x + 2)/(2x - 1))]^(((2x - 1)x)/(x^2 - 4x + 2))`
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 - 4x + 2))^x = [lim_(x -> oo) "e"]^((2x^2 - x)/(x^2 - 4x + 2))`
`lim_(x -> oo) (1 + 1/x)^x` = e
= `"e"^(lim_(x ->oo)) ((2x^2 - x)/(x^2 - 4x + 2))`
= `"e"^(lim_(x -> oo) (x^2(2 -x/x^2))/(x^2(1 - (4x)/(x^2) + 2/x^2))`
= `"e"^(lim_(x ->oo) ((2 - 1/x)/(1 -4/x + 2/x^2))`
= `"e"^(((2 - 0)/(1 - 0 + 0))`
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x` = e2
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