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Mole fraction and activity coefficient of component $1$ in a binary liquid mixture are $x_1$ and $\gamma _1$ respectively. $G^E$ is excess molar Gibbs energy of the mixture, $R$ is universal gas constant and $T$ is absolute temperature of the mixture. Which one of the following is always true?

- $\underset{x_1 \rightarrow 1}{\lim} \dfrac{G^E}{RT} =0 \\$
- $\underset{x_1 \rightarrow 1}{\lim} \dfrac{G^E}{RT} =0.5 \\$
- $\underset{x_1 \rightarrow 1}{\lim} \gamma_1 =0 \\$
- $\underset{x_1 \rightarrow 1}{\lim} \gamma_1 =0.5$