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Case 4: Each face is an \(n\)-gon with \(n \ge 6\text{.}\) Following the same procedure as above, we deduce that

\begin{equation*} k = \frac{2nf}{4+(n-2)f}\text{,} \end{equation*}

which will be increasing to a horizontal asymptote of \(\frac{2n}{n-2}\text{.}\) When \(n = 6\text{,}\) this asymptote is at \(k = 3\text{.}\) Any larger value of \(n\) will give an even smaller asymptote. Therefore no regular polyhedra exist with faces larger than pentagons. 8 

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