Statement 1: `lim_(xto oo)(1/(x^(2))+2/(x^(2))+3/(x^(2))+…………..+x/(x^(2)))=lim_(xto oo)1/(x^(2))+………….+lim_(xto oo)x/(x^(2))=0`
Statement 2: `lim_(xtoa)(f_(1)(x)+f_(2)(x)+………..+f_(n)(x))=lim_(xtoa)f_(1)(x)+………….+lim_(xtoa)f_(n)(x)` provided each limit exists individually.

Options

A. Statement-1 is True, Statement -2 is True, Statement -2 is True and Statement -2 is a correct explanation for statement -1.
B. Statement -1 is True, Statement -2 is True and Statement -2 is Not is a correct explanation for Statement -1.
C. Statement -1 is True, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Answer

To solve the problem, we need to analyze both statements given in the question.

### Statement 1:
We need to evaluate the limit:
\[ \lim_{x \to \infty} \left( \frac{1}{x^2} + \frac{2}{x^2} + \frac{3}{x^2} + \ldots + \frac{x}{x^2} \right) \]

This can be rewritten as:
\[ \lim_{x \to \infty} \left( \frac{1 + 2 + 3 + \ldots + x}{x^2} \right) \]

The sum of the first \( n \) natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]

So, for \( n = x \):
\[ 1 + 2 + 3 + \ldots + x = \frac{x(x + 1)}{2} \]

Substituting this back into our limit, we have:
\[ \lim_{x \to \infty} \frac{\frac{x(x + 1)}{2}}{x^2} \]

This simplifies to:
\[ \lim_{x \to \infty} \frac{x(x + 1)}{2x^2} = \lim_{x \to \infty} \frac{x + 1}{2x} = \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{2} \]

As \( x \to \infty \), \( \frac{1}{x} \to 0 \):
\[ \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{2} = \frac{1 + 0}{2} = \frac{1}{2} \]

Thus, the limit evaluates to:
\[ \frac{1}{2} \]

### Statement 2:
The second statement states:
\[ \lim_{x \to a} (f_1(x) + f_2(x) + \ldots + f_n(x)) = \lim_{x \to a} f_1(x) + \lim_{x \to a} f_2(x) + \ldots + \lim_{x \to a} f_n(x) \]
provided each limit exists individually.

This is a well-known property of limits, known as the linearity of limits. Therefore, if each individual limit exists, the limit of the sum is equal to the sum of the limits.

### Conclusion:
- Statement 1 is **incorrect**; the limit evaluates to \( \frac{1}{2} \) not \( 0 \).
- Statement 2 is **correct**.

### Final Answer:
- Statement 1: **False**
- Statement 2: **True**