Question

Evaluate the following Limits: `lim_(x -> 0)((1 - x)^5 - 1)/((1 - x)^3 - 1)`

Answer 1

`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`