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Question
Evaluate the following Limits: `lim_(x -> 0)((1 - x)^5 - 1)/((1 - x)^3 - 1)`
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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))/((y^3 - 1)/(y - 1))) ...[("As" y -> 1"," y ≠ 1),(therefore y - 1 ≠0),("Divide Numerator and"),("Denominator by " y - 1)]`
= `(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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