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Question
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
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Solution
Given:
\[f\left( x \right) = \binom{\frac{x^2 - 3x + 2}{x - 1}, if x \neq 1}{k, if x = 1}\]
If
\[f\left( x \right)\]is continuous at x = 1, then,
\[\lim_{x \to 1} f\left( x \right) = f\left( 1 \right)\]
\[\lim_{x \to 1} \frac{x^2 - 3x + 2}{x - 1} = k\]
\[\lim_{x \to 1} \frac{\left( x - 2 \right)\left( x - 1 \right)}{x - 1} = k\]
\[\lim_{x \to 1} \left( x - 2 \right) = k\]
\[k = - 1\]
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