Question

\(\lim_{x\rightarrow \infty}\){\(x-\sqrt[n]{(x-a_1)(x-a_2)......(x-a_n)}\)} where a₁,a₂,.....a_n are positive rational numbers.The limit

• A. does not exist
• B. is \(\frac{a_1+a_2+....+a_n}{n}\)
• C. is \(\sqrt[n]{a_1a_2.....a_n}\)
• D. is \(\frac{n}{a_1+a_2+....+a_n}\)

Answer 1

The Correct Option is B

Given:
\[ \lim_{x \rightarrow \infty} \left( x - \sqrt[x]{(x - a_1)(x - a_2) \ldots (x - a_n)} \right) \]

As \( x \to \infty \), each \( x - a_i \approx x \), so:
\[ (x - a_1)(x - a_2) \ldots (x - a_n) \approx x^n \left( 1 - \frac{a_1}{x} \right)\left( 1 - \frac{a_2}{x} \right) \ldots \left( 1 - \frac{a_n}{x} \right) \]

So we get: 
\[ \sqrt[x]{(x - a_1)\ldots(x - a_n)} \approx x \cdot \sqrt[x]{\left( 1 - \frac{a_1}{x} \right) \ldots \left( 1 - \frac{a_n}{x} \right)} \]

Now using the approximation:
\[ \sqrt[x]{\left( 1 - \frac{a_1}{x} \right) \ldots \left( 1 - \frac{a_n}{x} \right)} \approx 1 - \frac{a_1 + a_2 + \ldots + a_n}{x} \]

So:
\[ x - \sqrt[x]{(x - a_1)\ldots(x - a_n)} \approx x - x \left( 1 - \frac{a_1 + a_2 + \ldots + a_n}{x} \right) = \frac{a_1 + a_2 + \ldots + a_n}{n} \]

Final Answer: Option (B): \( \frac{a_1 + a_2 + \ldots + a_n}{n} \)

Answer 2

We want to find the limit:

\(\lim_{x \to \infty} \left\{ x - \sqrt[n]{(x-a_1)(x-a_2) \cdots (x-a_n)} \right\}\)

where \(a_1, a_2, \dots, a_n\) are positive rational numbers. 

Let's rewrite the expression inside the limit:

\(x - \sqrt[n]{(x-a_1)(x-a_2) \cdots (x-a_n)} = x - \sqrt[n]{x^n - (a_1 + a_2 + \dots + a_n)x^{n-1} + \cdots + (-1)^n a_1 a_2 \dots a_n}\)

Let \(S = a_1 + a_2 + \dots + a_n\). Then the expression inside the root can be written as:

\(x^n - Sx^{n-1} + O(x^{n-2})\)

Let's factor out \(x^n\) from the root:

\(x - \sqrt[n]{x^n \left( 1 - \frac{S}{x} + O\left(\frac{1}{x^2}\right) \right)}\)

\(x - x \sqrt[n]{1 - \frac{S}{x} + O\left(\frac{1}{x^2}\right)}\)

Now we can use the binomial approximation: \((1+y)^k \approx 1 + ky\) for small y.

In our case, \(y = -\frac{S}{x} + O\left(\frac{1}{x^2}\right)\) and \(k = \frac{1}{n}\).

\(x - x \left( 1 + \frac{1}{n} \left( -\frac{S}{x} + O\left(\frac{1}{x^2}\right) \right) + O\left(\frac{1}{x^2}\right) \right)\)

\(x - x \left( 1 - \frac{S}{nx} + O\left(\frac{1}{x^2}\right) \right)\)

\(x - x + \frac{S}{n} + O\left(\frac{1}{x}\right)\)

\(\frac{S}{n} + O\left(\frac{1}{x}\right)\)

As \(x \to \infty\), the term \(O\left(\frac{1}{x}\right)\) goes to 0. So, we have:

\(\lim_{x \to \infty} \left\{ x - \sqrt[n]{(x-a_1)(x-a_2) \cdots (x-a_n)} \right\} = \frac{S}{n} = \frac{a_1 + a_2 + \dots + a_n}{n}\)

Conclusion:

The limit is \(\frac{a_1 + a_2 + \dots + a_n}{n}\)