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प्रश्न
Evaluate the following :
`lim_(x -> ∞) [((2x - 1)^20 (3x - 1)^30)/(2x + 1)^50]`
उत्तर
Let L = `lim_(x -> ∞) [((2x - 1)^20 (3x - 1)^30)/(2x + 1)^50]`
Dividing numerator and denominator by x50, we get,
L = `lim_(x -> ∞) ((2x - 1)^20/x^20 xx (3x - 1)^30/x^30)/((2x + 1)^50/x^50`
= `lim_(x -> ∞) (((2x - 1)/x)^20 xx ((3x - 1)/x)^30)/((2x + 1)/x)^50`
= `lim_(x -> ∞) ((2 - 1/x)^20 xx (3 - 1/x)^30)/(2 + 1/x)^50`
= `([lim_(x -> ∞) (2 - 1/x)^20] xx [lim_(x -> ∞) (3 - 1/x)^30])/(lim_(x ->∞) (2 + 1/x)^50`
= `((2 - 0)^20 (3 - 0)^30)/(2 + 0)^50 ...[because lim_(x -> ∞) 1/x = 0]`
= `(2^20 xx 3^30)/(2^50)`
= `(3/2)^30`.
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