Going to the Hospital

Calculus Level 4

Let f(x)=xe2f(x) = x e^2 and g(x)=xlnx,g(x) = x^{\ln x}, and let α\alpha and β\beta with α<β\alpha < \beta be the two roots of f(x)g(x)=0.f(x)-g(x) = 0. Also, let

l=limxβ f(x)cβg(x)β2. l= \lim_{x \rightarrow \beta} \ \dfrac{f(x)-c\beta}{g(x)-\beta^2}.

Then what is the value of cl?c-l?

Note: You may use the approximation e2.7183. e \approx 2.7183 .


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