Question

How do I prove that the limit of this function is `1/n`? (`n` is a positive integer)

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

Answer 1

We want to evaluate the limit:

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

If we put `x=0` we can see that we have a limit of an indeterminate form, so we are able to apply L'Hôpital's rule, which states that if a limit:

`lim_(x rarr a) f(x)/g(x)`

is of an indeterminate form `0/0`, then, providing the limit exists:

`lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f'(x))/(g'(x))`

Hence, we have (as `n` is a constant):

`L = lim_(x->0) (1/n(1+x)^(1/n-1)-0)/1`
`\ \ = lim_(x->0) 1/n(1+x)^(1/n-1)`
`\ \ = 1/n \ lim_(x->0) (1+x)^(1/n-1)`

`\ \ = 1/n * 1`

`\ \ = 1/n` QED