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:** You may use the approximation $e \approx 2.7183 .$

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