\(\lim_{x\rightarrow \infty}\){\(x-\sqrt[n]{(x-a_1)(x-a_2)......(x-a_n)}\)} where a1,a2,.....an are positive rational numbers.The limit
- does not exist
is \(\frac{a_1+a_2+....+a_n}{n}\)
- is \(\sqrt[n]{a_1a_2.....a_n}\)
- is \(\frac{n}{a_1+a_2+....+a_n}\)
The Correct Option is B
Approach Solution - 1
The correct option is(B): \(\frac{a_1+a_2+....+a_n}{n}\)
Approach Solution -2
Given:
\(\lim_{x \rightarrow \infty} \left( x - \sqrt[x]{(x-a_1)(x-a_2) \ldots (x-a_n)} \right)\)
As x becomes very large, each \(x - a_i\) where \(a_i\) are constants becomes negligible compared to x. Therefore, we can approximate:
\(\sqrt[x]{(x-a_1)(x-a_2) \ldots (x-a_n)} \approx \sqrt[x]{x^n \left( 1 - \frac{a_1}{x} \right)\left( 1 - \frac{a_2}{x} \right) \ldots \left( 1 - \frac{a_n}{x} \right)}\)
\(\approx x \left[ 1 - \left( 1 - \frac{a_1}{x} \right)^{1/n} \left( 1 - \frac{a_2}{x} \right)^{1/n} \ldots \left( 1 - \frac{a_n}{x} \right)^{1/n} \right]\)
\(\approx x \left[ 1 - \left( 1 - \frac{a_1}{nx} \right)\left( 1 - \frac{a_2}{nx} \right) \ldots \left( 1 - \frac{a_n}{nx} \right) \right]\)
\(\approx x \left[ \frac{a_1 + a_2 + \ldots + a_n}{nx} \right]\)
\(= \frac{a_1 + a_2 + \ldots + a_n}{n}\)
So, the correct option (B): \(\frac{a_1 + a_2 + \ldots + a_n}{n}\).
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).
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