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Question
Evaluate the following :
`lim_(x -> ∞) [(x^3 + 3x + 2)/((x + 4)(x - 6)(x - 3))]`
Solution
Let L = `lim_(x -> ∞) [(x^3 + 3x + 2)/((x + 4)(x - 6)(x - 3))]`
Dividing numerator and denominator by x3, we get,
L = `lim_(x -> ∞) (1 + 3/x^2 + 2/x^3)/(((x + 4)/x)((x - 6)/x)((x - 3)/x)`
= `lim_(x -> ∞) (1 + 3/x^2 + 2/x^3)/((1 + 4/x)(1 - 6/x)(1 - 3/x))`
= `(lim_(x -> ∞)[1 + 3 xx 1/x^2 + 2 xx 1/x^3])/([lim_(x -> ∞) (1 + 4 xx 1/x)] xx [lim_(x -> ∞) (1 - 6 xx 1/x)] xx [lim_(x -> ∞) (1 - 3 xx 1/x)])`
= `(1 + 3 xx 0 + 2 xx 0)/([1 + 4 xx 0] xx [1 - 6 xx 0] xx [ 1 - 3 xx 0]) ...[because lim_(x -> ∞) 1/x^"n" = 0 "if n" > 0]`
= 1
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