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A positive element $ a $ in a $C^*$-algebra $ A $ has the property that for every $ f \in A^{*}_{+} $ (note that $ A^{*}_{+} $ denotes the set of all positive linear functionals) and $ f \neq 0 $, we have $ f(a) \geq 0 $. Please show that

$ \displaystyle \lim_{\epsilon \to 0^{+}} \| {g_{\epsilon}}(a) x - x \| = 0 $ for all $ x \in A $, where $ {g_{\epsilon}}(t) := \dfrac{t}{t + \epsilon} $ for $ \epsilon > 0 $.

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