Question

Let $a_1, a_2, a_3, \ldots \ldots \ldots, a_n$ be fixed real numbers and define a function $f(x)=\left(x-a_1\right)\left(x-a_2\right) \ldots \ldots\left(x-a_n\right)$, then

• A. $\displaystyle\lim _{x \rightarrow a_1} f(x)=0$
• B. $\displaystyle\lim _{x \rightarrow a} f(x)=\left(a-a_1\right)\left(a-a_2\right) \ldots \ldots . .\left(a-a_n\right)$, for some $a \neq a_1$,
• C. Both (a) and (b) are true
• D. Either (a) or (b) is true

Answer 1

Answer: (C)

Given, $f(x)=\left(x-a_1\right)\left(x-a_2\right) \ldots .\left(x-a_n\right)$
$\therefore \displaystyle\lim _{x \rightarrow a_1} f(x) =\displaystyle\lim _{x \rightarrow a_1}\left(x-a_1\right)\left(x-a_2\right) \ldots\left(x-a_n\right) $
$ =\left(a_1-a_1\right)\left(a_1-a_2\right) \ldots .\left(a_1-a_n\right) $
$ =0 \times\left(a_1-a_2\right) \ldots\left(a_1-a_n\right)=0$
Again, $\displaystyle\lim _{x \rightarrow a} f(x)=\displaystyle\lim _{x \rightarrow a}\left(x-a_1\right)\left(x-a_2\right) \ldots\left(x-a_n\right)$
$=\left(a-a_1\right)\left(a-a_2\right) \ldots\left(a-a_n\right)$