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Question
Evaluate the following limit :
`lim_(x -> 0)[((1 - x)^8 - 1)/((1 - x)^2 - 1)]`
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Solution
Put 1 – x = y
Then as x → 0, y → 1
∴ `lim_(x -> 0) ((1 - x)^8 - 1)/((1 - x)^2 - 1)`
= `lim_(y -> 1) (y^8 - 1)/(y^2 - 1)`
= `lim_(y -> 1)[(((y^8 - 1)/(y - 1)))/(((y^2 - 1)/(y - 1)))]` ...[∵ y → 1, y ≠ 1, ∴ y – 1 ≠ 0]
= `(lim_(y -> 1)((y^8 - 1)/(y - 1)))/(lim_(y -> 1)((y^2 - 1)/(y - 1))`
= `(8(1)^7)/(2(1)^1) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= 4
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