Let \(f(x) = x e^2\) and \(g(x) = x^{\ln x},\) and let \(\alpha\) and \(\beta\) with \(\alpha < \beta\) be the two roots of \(f(x)-g(x) = 0.\) Also, let

\[ l= \lim_{x \rightarrow \beta} \ \dfrac{f(x)-c\beta}{g(x)-\beta^2}. \]

Then what is the value of \(c-l?\)

**Note:** Try my set.

**Details and Assumptions**

You may use the approximation \( e \approx 2.7183 .\)

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