Limit of $\frac{x^y-y^x}{x^x-y^y} $ while $x\to y$
$$\lim_{x\to y}\frac{x^y-y^x}{x^x-y^y} $$ I've tried L'Hospital and this. Both doesn't tends to work. Please help someone.
Solution 1:
L'Hôspital's rule works: $$\lim_{x \to y} \frac{x^y-y^x}{x^x-y^y} = \lim_{x \to y} \frac{yx^{y-1}-y^x \ln y}{x^x \ln x +x^x} = \frac{1-\ln y}{1+\ln y}.$$