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Question
Evaluate the following :
`lim_(x -> 0)[((1 - x)^5 - 1)/((1 - x)^3 - 1)]`
Solution
`lim_(x -> 0)[((1 - x)^5 - 1)/((1 - x)^3 - 1)]`
Put 1 – x = y
As x → 0, y → 1
∴ `lim_(x -> 0)[((1 - x)^5 - 1)/((1 - x)^3 - 1)]`
= `lim_(y -> 1) (y^5 - 1)/(y^3 - 1)`
= `(lim_(y -> 1) (y^5 - 1)/(y - 1))/(lim_(y -> 1)(y^3 - 1)/(y - 1))` ...[∵ y → 1, ∴ y ≠ 1, ∴ y – 1 ≠ 0]
= `(lim_(y -> 1)(y^5 - 1^5)/(y - 1))/(lim_(y -> 1)(y^3 - 1^3)/(y - 1))`
= `(5(1)^4)/(3(1)^2) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `5/3`
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