If f(x) = lim n tends to infinity 2/n^2 (summation k=1 to n (kx))(3^nx-1/3^nx+1) , then find sum of all the solutions of the equations f(x)=|x^2-2|

To tackle the problem, we need to analyze the function \( f(x) \) defined as follows:

\[ f(x) = \lim_{n \to \infty} \frac{2}{n^2} \left( \sum_{k=1}^{n} kx \right) \left( \frac{3^n x - 1}{3^n x + 1} \right) \]

First, let's break this down step by step.

Step 1: Simplifying the Summation

The summation \( \sum_{k=1}^{n} kx \) can be simplified using the formula for the sum of the first \( n \) natural numbers:

\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]

Thus, we can rewrite the summation as:

\[ \sum_{k=1}^{n} kx = x \cdot \frac{n(n+1)}{2} \]

Step 2: Substituting Back into \( f(x) \)

Now substituting this back into our expression for \( f(x) \), we have:

\[ f(x) = \lim_{n \to \infty} \frac{2}{n^2} \left( x \cdot \frac{n(n+1)}{2} \right) \left( \frac{3^n x - 1}{3^n x + 1} \right) \]

This simplifies to:

\[ f(x) = \lim_{n \to \infty} \frac{x(n^2 + n)}{n^2} \left( \frac{3^n x - 1}{3^n x + 1} \right) \]

As \( n \to \infty \), the term \( \frac{n^2 + n}{n^2} \) approaches 1. Therefore, we can focus on the limit of the remaining expression:

Step 3: Analyzing the Limit

Next, we need to evaluate:

\[ \lim_{n \to \infty} \left( \frac{3^n x - 1}{3^n x + 1} \right) \]

For \( x \neq 0 \), as \( n \to \infty \), this limit approaches 1, since the \( 3^n x \) term dominates both the numerator and the denominator. Thus, we find:

\[ f(x) = x \cdot 1 = x \]

Step 4: Setting Up the Equation

Now we need to solve the equation:

\[ f(x) = |x^2 - 2| \]

This translates to:

\[ x = |x^2 - 2| \]

Step 5: Solving the Absolute Value Equation

We can break this into two cases based on the definition of absolute value:

Case 1: \( x^2 - 2 \geq 0 \)

In this case, we have:

\[ x = x^2 - 2 \]

Rearranging gives:

\[ x^2 - x - 2 = 0 \]

This factors to:

\[ (x - 2)(x + 1) = 0 \]

Thus, the solutions are:

\[ x = 2 \quad \text{and} \quad x = -1 \]

Case 2: \( x^2 - 2 < 0 \)

Here, we have:

\[ x = -(x^2 - 2) \]

Rearranging gives:

\[ x^2 + x - 2 = 0 \]

This factors to:

\[ (x - 1)(x + 2) = 0 \]

Thus, the solutions are:

\[ x = 1 \quad \text{and} \quad x = -2 \]

Step 6: Summing All Solutions

Now we have four solutions: \( 2, -1, 1, -2 \). To find the sum of all solutions:

\[ 2 + (-1) + 1 + (-2) = 0 \]

Therefore, the sum of all the solutions to the equation \( f(x) = |x^2 - 2| \) is:

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