Let a1, a2, ....,an be fixed real numbers and define a function f… - Vidyadip

Question

Let a₁, a₂, ....,a_n be fixed real numbers and define a function f(x) = (x - a₁)(x - a₂)... (x - a_n). What is & $\mathop {\lim }\limits_{x \to {a_1}} f(x)$? For some $a \ne {a_1},\;{a_2}...{a_n},$ compute $\mathop {\lim }\limits_{x \to {a}} f(x)$

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Answer

Here f(x) = (x – a₁) (x – a₂)… (x – a_n)
Now $\mathop {\lim }\limits_{x \to {a_1}} f(x) = \mathop {\lim }\limits_{x \to {a_1}} (x - {a_1})(x - {a_2})...(x - {a_n})$
= (a₁ – a₁) (a₁ – a₂) … (a₁ – a_n)
= 0 × (a₁ – a₂) … (a₁ – a_n) = 0.
Also $\mathop {\lim }\limits_{x \to a} f(x) = \mathop {\lim }\limits_{x \to a} (x - {a_1})(x - {a_2})...(x - {a_n})$
= (a – a₁) (a – a₂) … (a – a_n)

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