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