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Question
Let a1, a2, ..., an be fixed real numbers such that
f(x) = (x − a1) (x − a2) ... (x − an)
What is \[\lim_{x \to a_1} f\left( x \right)?\] Compute \[\lim_{x \to a} f\left( x \right) .\]
Solution
\[f\left( x \right) = \left( x - a_1 \right) \left( x - a_2 \right) . . . \left( x - a_n \right)\]
\[ \lim_{x \to a_1} f\left( x \right)\]
\[ = \lim_{x \to a_1} \left[ \left( x - a_1 \right) \left( x - a_2 \right) . . . \left( x - a_n \right) \right]\]
\[ = \left( a_1 - a_1 \right) \left( a_1 - a_2 \right) . . . \left( a_1 - a_n \right)\]
\[ = 0\]
\[ \lim_{x \to a} f\left( x \right)\]
\[ = \lim_{x \to a} \left( x - a_1 \right) \left( x - a_2 \right) . . . \left( x - a_n \right)\]
\[ = \left( a - a_1 \right) \left( a - a_2 \right) . . . \left( a - a_n \right)\]
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