Question

Limited x-0(1+nx)1/x=?

Answer 1

`e^n`

Explanation:

Plugging in value of `x`, this is `1^oo` form.
As any number `a` can be written as `e^lna`

`lim_(xto0) e^(ln(1+nx)^(1/x)`

`lim_(xto0) e^((1/x)(ln(1+nx))`

Multiplying and dividing by `n` to use standard limit :
`lim_(xto0) ln(1+x)/x =1`

`lim_(xto0) e^((1/(nx))(ln(1+nx)n)`

`lim_(xto0) e^(ln(1+nx)/(nx)n`

Limit can be taken inside the exponential.

`e^(lim_(xto0)(ln(1+nx)/(nx)n)`

The limit is just `1`

`e^n`