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Calculus Level 5

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.
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