Let \( f(x)\left\{\begin{array}{c}\int_{-1}^{x}|t-2| d t ; \quad x \neq 2 \\ k ; \quad x=2\end{a...

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Let \( f(x)\left\{\begin{array}{c}\int_{-1}^{x}|t-2| d t ; \quad x \neq 2 \\ k ; \quad x=2\end{array}\right. \). If \( f(x) \) is continuous at \( x=2 \), then the value of \( k \) is equal to:
(a) \( \frac{3}{2} \)
(b) \( \frac{5}{2} \)
(c) \( \frac{9}{2} \)
(d) \( \frac{7}{2} \)
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