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Question
Evaluate the following :
`lim_(x -> ∞) [((3x - 4)^3 (4x + 3)^4)/(3x + 2)^7]`
Solution
Let L = `lim_(x -> ∞) [((3x - 4)^3 (4x + 3)^4)/(3x + 2)^7]`
Dividing numerator and denominator by x7, we get,
L = `lim_(x -> ∞) ((3x - 4)^3/x^3 xx (4x + 3)^4/x^4)/((3x + 2)^7/x^7)`
= `lim_(x -> ∞) (((3x - 4)/x)^3 xx ((4x + 3)/x)^4)/((3x + 2)/x)^7`
= `lim_(x -> ∞) ((3 - 4/x)^3 xx (4 + 3/x)^4)/(3 + 2/x)^7`
= `([lim_(x -> ∞) (3 - 4 xx 1/x)^3] xx [lim_(x ->∞) (4 + 3 xx 1/x)^4])/(lim_(x ->∞) (3 + 2 xx 1/x)^7`
= `((3 - 4 xx 0)^3 xx (4 + 3 xx 0)^4)/((3 + 2 xx 0)^7) ...[because lim_(x -> ∞) 1/x = 0]`
= `(3^3 xx 4^4)/(3^7)`
= `256/81`.
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