Let a1- a2- an be fixed real numbers and define a function fx-x-a1x-a2-x-anWhat is lim x - a1 fx - For some a - a1-a2- an compute lim x - a fx-

Name: NCERT - Grade 11 - Mathematics - Limits and Derivatives - Q33
Uploaded: 2022-07-04T10:36:42+05:30
Description: Given : nbsp;f(x)=(x−a_1)(x−a_2)⋯(x−a_n) nbsp;lim_ x → a_1 f(x) = nbsp; nbsp;lim_ x → a_1(x−a_1)(x−a_2)⋯(x−a_n) =(a_1−a_1)(a_1−a_2)⋯(a_1−a_n) =0 × (a_1−a_ ...

Given : nbsp;f(x)=(x−a_1)(x−a_2)⋯(x−a_n) nbsp;lim_ x → a_1 f(x) = nbsp; nbsp;lim_ x → a_1(x−a_1)(x−a_2)⋯(x−a_n) =(a_1−a_1)(a_1−a_2)⋯(a_1−a_n) =0 × (a_1−a_ ...

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Byju's Answer

Standard XII

Mathematics

Graph of [x]

Let a1, a2,⋯,...

Question

Let $a_{1}, a_{2}, \dots, a_{n}$ be fixed real numbers and define a function $f (x) = (x - a_{1}) (x - a_{2}) \dots (x - a_{n})$
What is $lim x \to a_{1} f (x) ?$ For some $a \neq a_{1}, a_{2} \dots . a_{n}$ compute $lim x \to a f (x)$ .

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Solution

Given : $f (x) = (x - a_{1}) (x - a_{2}) \dots (x - a_{n})$
$lim x \to a_{1} f (x) = lim x \to a_{1} (x - a_{1}) (x - a_{2}) \dots (x - a_{n})$
$= (a_{1} - a_{1}) (a_{1} - a_{2}) \dots (a_{1} - a_{n})$
$= 0 \times (a_{1} - a_{2}) \dots (a_{1} - a_{n})$
$= 0$
Hence, $lim x \to a_{1} f (x) = 0$
Now,
$lim x \to a f (x) = lim x \to a (x - a_{1}) (x - a_{2}) \dots (x - a_{n})$
Substituting $x = a$
$= (a - a_{1}) (a - a_{2}) \dots (a - a_{n})$
Hence, $lim x \to a f (x) = (a - a_{1}) (a - a_{2}) \dots (a - a_{n})$

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