\textbf{117.} If $\theta$ is the angle between the left and right tangents to the graph of the function $f(x) = \left[x + \frac{1}{2}\right]x + x^2$, at the point $x = \frac{1}{2}$, what is $\tan\theta$?
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(1) $\dfrac{1}{4}$ \hspace{2cm} (2) $\dfrac{1}{2}$ \hspace{2cm} (3) $\dfrac{2}{3}$ \hspace{2cm} (4) $\dfrac{3}{4}$
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