Let a 1- a 2- - - a n be fixed real numbers and let fx-x-a1x-a2x-a3 -x-anFind lim n fxIf q-a1- a2- - a n- compute limfx - ax

X→ a_1^+limf(x)=h→ 0lim f(a_1+h) h→ 0lim|(a_1+h a_1)(a_1+h a_2)(a_1+h a_3)...(a_1+h a_n)| =h→ 0lim|h(a_1+h a_2)(a_1+h a_3)...(a_1+h a_n)| =(0× (a_1 a_2)× (a_1 a ...

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Byju's Answer

Standard XI

Mathematics

Theorems for Differentiability

Let a 1, a 2,...

Question

Let $a_{1}, a_{2}, . . . . a_{n}$ be fixed real numbers and let

$f (x) = (x - a_{1}) (x - a_{2}) (x - a_{3}) . . . (x - a_{n})$ .

Find $l i m x \to a_{1} f (x),$

If $a \neq a_{1}, a_{2}, . . . a_{n}, c o m p u t e l i m x \to a f (x)$

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Solution

$l i m x \to a_{1}^{+} f (x) = l i m h \to 0 f (a_{1} + h)$

$l i m h \to 0 | (a_{1} + h - a_{1}) (a_{1} + h - a_{2}) (a_{1} + h - a_{3}) . . . (a_{1} + h - a_{n}) |$

$= l i m h \to 0 | h (a_{1} + h - a_{2}) (a_{1} + h - a_{3}) . . . (a_{1} + h - a_{n}) |$

$= (0 \times (a_{1} - a_{2}) \times (a_{1} - a_{3}) \times . . . \times (a_{1} - a_{n}) = 0.$

$l i m x \to a_{1}^{-} f (x) = l i m h \to 0 f (a_{1} - h)$

$= l i m h \to 0 {(a_{1} - h - a_{1}) (a_{1} - h - a_{2}) (a_{1} - h - a_{3}) . . . (a_{1} - h - a_{n})}$

$= l i m h \to 0 {(- h) (a_{1} - h - a_{2}) (a_{1} - h - a_{3}) . . . (a_{1} - h - a_{n})}$

$= {0 \times (a_{1} - a_{2}) \times (a_{1} - h - a_{3}) \times . . . \times (a_{1} - a_{n})} = 0.$

$∴ l i m x \to a_{1}^{+} f (x) = l i m x \to a_{1}^{-} f (x) = 0.$

$H e n c e l i m x \to a_{1}$

$F o r a n y a \neq a_{1}, a_{2},,,, a_{n}, w e h a v e$

$l i m x \to a f (x) = l i m x \to a {(x - a_{1}) (x - a_{2}) (x - a_{3}) . . . (x - a_{n}) .}$

$= (a - a_{1}) (a - a_{2}) (a - a_{3}) . . . (x - a_{n}) .$

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Similar questions

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