Shaped into a box

Suppose we define a type of function $f\left( x \right) =\dfrac { k{ x }^{ n } }{ n+{ x }^{ n } }$ with $n>0$ and $k>0$ . To calculate a convergent area above the curve for $f\left( x \right)$ over the domain of all real numbers, one can bound this area with a horizontal line, $\displaystyle y=\lim _{ x\rightarrow \pm \infty }{ f\left( x \right) }$ .

Using this information, find an expression for this area as $\displaystyle \lim _{ n\rightarrow \infty }{ f\left( x \right) }$ . Write your answer in terms of $k$ .

This problem is original.