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Question
Evaluate the following :
`lim_(x -> ∞) [(8x^2 + 5x + 3)/(2x^2 - 7x - 5)]^((4x + 3)/(8x - 1))`
Solution
`lim_(x -> ∞) [(8x^2 + 5x + 3)/(2x^2 - 7x - 5)]^((4x + 3)/(8x - 1))`
= `[lim_(x -> ∞) ((8x^2 + 5x + 3)/(2x^2 - 7x - 5))]^(lim_(x -> ∞)(4x + 3)/(8x - 1))`
Consider
`lim_(x -> ∞) (8x^2 + 5x + 3)/(2x^2 - 7x - 5)`
= `lim_(x -> ∞) [((8x^2 + 5x + 3)/x^2)/((2x^2 - 7x - 5)/x^2)] ...[("Divide Numerator and"),("Denominator by" x^2)]`
= `(lim_(x -> ∞)(8 + 5/x + 3/x^2))/(lim_(x -> ∞)(2 - 7/x - 5/x^2))`
= `(8 + 0 + 0)/(2 - 0 - 0) ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= 4
Consider
`lim_(x -> ∞) (4x + 3)/(8x - 1)`
= `lim_(x -> ∞) ((4x + 3)/x)/((8x - 1)/x)` ...[Divide Numerator and Denominator by x]
= `(lim_(x -> ∞)(4 + 3/x))/(lim_(x -> ∞)(8 - 1/x)`
= `(4 + 0)/(8 - 0) ...[lim_(x -> ∞) 1/x = 0]`
= `1/2`
Required limit = `(4)^(1/2)` = 2
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