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Question
Evaluate the following :
`lim_(x -> ∞) [((3x^2 + 4)(4x^2 - 6)(5x^2 + 2))/(4x^6 + 2x^4 - 1)]`
Solution
`lim_(x -> ∞) (((3x^2 + 4)(4x^2 - 6)(5x^2 + 2))/x^6)/((4x^6 + 2x^4 - 1)/x^6)`
= `lim_(x -> ∞) (((3x^2 + 4)(4x^2 - 6)(5x^2 + 2))/(x^6))/((4x^6 + 2x^4 - 1)/(x^6)) ...[("Divide numerator and"),("denominator by" x^6)]`
= `lim_(x -> ∞) (((3x^2 + 4)/x^2)((4x^2 - 6)/x^2)((5x^2 + 2)/x^2))/(4 + 2/x^2 - 1/x^6)`
= `(lim_(x -> ∞) (3 + 4/x^2)(4 - 6/x^2)(5 + 2/x^2))/(lim_(x -> ∞)(4 + 2/x^2 - 1/x^6)`
= `(lim_(x -> ∞) (3 + 4/x^2)*lim_(x -> ∞) 4(4 - 6/x^2)*lim_(x -> ∞) (5 + 2/x^2))/(lim_(x -> ∞)4 + lim_(x -> ∞) 2/x^2 - lim_(x -> ∞)1/x^6)`
= `((3 + 0)(4 - 0)(5 + 0))/(4 + 0 - 0) ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= `(3 xx 4 xx 5)/4`
= 15
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