Let $x _ { 0 }$ be the real number such that $e ^ { x _ { 0 } } + x _ { 0 } = 0$. For a given real number $\alpha$, define

$$g ( x ) = \frac { 3 x e ^ { x } + 3 x - \alpha e ^ { x } - \alpha x } { 3 \left( e ^ { x } + 1 \right) }$$

for all real numbers $x$.

Then which one of the following statements is TRUE?

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(A) & For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 0$ \\
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(B) & For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 1$ \\
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(C) & For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 0$ \\
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(D) & For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = \frac { 2 } { 3 }$ \\
\hline
\end{tabular}
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