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