Let a1, a2,..., an be fixed real numbers and define a function f ( x) = ( x − a1 ) ( x − a2 )...( x − an ). What is limx→a1f(x) ? For some a ≠ a1, a2, ..., an, compute limx→af(x) - Mathematics

प्रश्न

Let a₁, a₂,..., a_nbe fixed real numbers and define a function f ( x) = ( x − a₁ ) ( x − a₂ )...( x − a_n ).

What is $lim_{x \to a_{1}} f (x)$ ? For some a ≠ a₁, a₂, ..., a_n, compute $lim_{x \to a} f (x)$

बेरीज

उत्तर

To factor (x − a₁),

If x → a₁, x − a₁ → 0

∴ $lim_{x - a_{1}} (x - a_{2}) = (a_{1} - a_{2})$

f(x) = (x – a₁) (x – a₂) ….. (x – a_n)

$lim_{x - a_{1}} f (x) = lim_{x - a_{1}}$ (x – a₁) (x – a₂) ….. (x – a_n)

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

= 0 × (a₁ − a₂) (a₁ − a₃) .......(a − a_n) 0

When, a ≠ a₁, a₂ ........, a_n

As soon as x → a, x → a₁ → a − a₁

a − a₁ is neither zero nor undefined.

Thus, the values of the second factor will be a − a₂, a − a₃ ....., a - a_n.

Hence, $lim_{x - a} f (x) = lim_{x - a}$ (x – a₁) (x – a₂) ….. (x – a_n)

= (a – a₁) (a – a₂) ….. (a – a_n)

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Concept of Limits - Limits of Polynomials and Rational Functions

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Q 29 Q 28 Q 30

पाठ 13: Limits and Derivatives - Exercise 13.1 [पृष्ठ ३०३]

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पाठ 13 Limits and Derivatives
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Let a1, a2,..., an be fixed real numbers and define a function f ( x) = ( x − a1 ) ( x − a2 )...( x − an ). What is limx→a1f(x) ? For some a ≠ a1, a2, ..., an, compute limx→af(x) - Mathematics

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