What is $lim_(x->0) ((1+x)^n-1)/x$ ?

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Answer 1 · 2017-11-04T11:59:04

$lim_(x->0) ((1+x)^n-1)/x =n$

If we assume the question was: $lim_(x->0) ((1+x)^n-1)/x$

We have an indeterminate limit of the form $0/0$

Hence, L'Hopital's rule applies.

$:.lim_(x->0) ((1+x)^n-1)/x = lim_(x->0) (d/dx((1+x)^n-1))/(d/dx(x))$

$= lim_(x->0) (n(1+x)^(n-1)xx1 -0)/1$ [Power rule and chain rule]

$= lim_(x->0) n(1+x)^(n-1)$

$= nxx1^(n-1)= n$

What is lim_(x->0) ((1+x)^n-1)/xlim_(x->0) ((1+x)^n-1)/x ?