Let $A$ be a unital C*-algebra. Show that an element $x$ of $A$ is self-adjoint if and only if $\lim_{t\to 0}\frac{1}{t}(||1+itx||-1)$=0.
My attempt: Suppose $x=x^*$. By functional calculus of x, there is the continuous function $f(\lambda)=1+it\lambda$ in $C(\sigma(x))$. Now $\lim_{t\to 0} \frac{1}{t}(||f||-1)=\lim_{t\to 0}\frac{1}{t}(\sup_{\lambda\in\sigma(x)}|1+it\lambda|-1)$. Because of compactness of $\sigma(x)$, there is $\mu\in\sigma(x)$ such that $||f||=|f(\mu)|$, thus,$$\lim_{t\to 0}\frac{1}{t}((1+t^2\mu^2)^{\frac{1}{2}}-1)=0$$ For converse direction, if $A$ is abelian, I can show it by using functional calculus of an arbitrary element and then easily I can conclude that this element is self adjoint. But now I can not use functional calculus, Please check my attempt and give me a hint for converse direction. Thanks in advance.