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Question

Let $$\displaystyle a_{1},\,a_{2},\,\cdot \cdot \cdot \cdot ,\,a_{n}$$ be fixed real numbers and define a function
$$\displaystyle f\left ( x \right )=\left ( x-a_{1} \right )\left ( x-a_{2} \right )$$.....$$\displaystyle \left ( x-a_{n} \right )$$
What is $$\displaystyle \lim_{x\rightarrow a_{1}}f\left ( x \right )?$$ For some $$\displaystyle a\neq a_{1},\,a_{2}\cdot \cdot \cdot ,\,a_{n},$$ compute $$\displaystyle \lim_{x\rightarrow a}f\left ( x \right )\cdot $$


Solution

We have $$\displaystyle f\left ( x \right )=\left (
x-a_{1} \right )(x-a_2).....\displaystyle \left ( x-a_n \right ) $$
$$\displaystyle
\lim_{x\rightarrow a_{1}}f\left ( x \right )=\lim_{x\rightarrow
a_{3}}\left [ \left ( x-a_{1} \right )\left ( x-a_{2} \right )\cdot
\cdot \cdot \left ( x-a_{n} \right ) \right ]$$
= $$\displaystyle
\left [ \lim_{x\rightarrow a_{1}}\left ( x-a_{1} \right ) \right ]\left [
\lim_{x\rightarrow a_{1}}\left ( x-a_{2} \right ) \right
]$$...$$\displaystyle \left [ \lim_{x\rightarrow a_{1}}\left ( x-a_{n}
\right ) \right ]$$
= $$\displaystyle \left ( a_{1}-a \right )\left ( a_{1}-a_{2} \right )$$...$$\displaystyle \left ( a_{1}-a _{n}\right )=0$$
$$\displaystyle \therefore \lim_{x\rightarrow a_{1}}f\left ( x \right )=0$$
Now $$\displaystyle
\lim_{x\rightarrow a}f\left ( x \right )=\lim_{x\rightarrow a}\left [
\left ( x-a_{1} \right )\left ( x-a_{2} \right )\cdot \cdot \cdot  \left
( x-a_{n} \right )\right ]$$
= $$\displaystyle \left [
\lim_{x\rightarrow a} \left ( x-a_{1} \right )\right ]\left [
\lim_{x\rightarrow a} \left ( x-a_{2} \right )\right
]$$...$$\displaystyle \left [ \lim_{x\rightarrow a} \left ( x-a
_{n}\right )\right ]$$
= $$\displaystyle \left ( a-a_{1} \right )\left ( a-a_{2} \right )$$...$$\displaystyle \left ( a-a_{n} \right )$$
$$\displaystyle
\therefore \lim_{x\rightarrow a}f\left ( x \right )=\left ( a-a_{1}
\right )\left ( a-a_{2} \right )$$...$$\displaystyle \left ( a-a_{n}
\right )$$

Mathematics
NCERT
Standard XI

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