Match the following List-I with List -II

List-I		List-II
(I)	If $f:R\to R$ is a differentiable function with property that  $\underset{x\to \infty }{\lim }f\left(x\right)=7$ and if $\underset{x\to \infty }{\lim }xf\text{'}\left(x\right)$ exists, then this limit is equal to	P	1
(II)	Let f and g be n-times continuously differentiable function in a neighbourhood of a point ‘a’ such that  $f\left(a\right)=g\left(a\right)=\ln 2$. $f^1\left(a\right)=g^1\left(a\right),\mathrm{......},f^{(n-1)}\left(a\right)=g^{(n-1)}\left(a\right)and{f}^n\left(a\right)\ne g^n\left(a\right),$ then  $\underset{x\to a}{\lim }\frac{e^{f\left(x\right)}-e^{g\left(x\right)}}{f\left(x\right)-g\left(x\right)}$ is	Q	3
(III)	Let $f:[a,b]\to R$ be a function continuous on [a,b] and differentiable on (a,b) if there exist $C\in (a,b)$ such that $\frac{f\left(b\right)-f\left(c\right)}{f\left(c\right)-f\left(a\right)}<0,$ then there always exist  $\mu \in (a,b),$ such that $f^1\left(\mu \right)$ is equal to	R	5
(IV)	Let a and b be positive real numbers, then $\underset{a}{\overset{b}{\int }}\frac{e^{\frac{x}{a}}-e^{\frac{b}{x}}}{x}dx$ is	S	7
		T	2
		U	0