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Question
`lim_(x -> 0) ((1 + x)^n - 1)/x` is equal to ______.
Options
n
1
– n
0
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Solution
`lim_(x -> 0) ((1 + x)^n - 1)/x` is equal to n.
Explanation:
Given `lim_(x -> 0) ((1 + x)^n - 1)/x`
= `lim_(x -> 0) ((1 + x)^n - (1)^n)/((1 + x) - (1))`
= `lim_(1 + x -> 1) ((1 + x)^n - (1)^n)/((1 + x) - (1))`
= `n(1)^(n - 1)`
= `n[lim_(x -> a) (x^n - a^n)/(x - a) = na^(n - 1)]`
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