For each n ≥ 1 define f_n : ℝ → ℝ by  x ∈ ℝ

where √ denotes the non-negative square root. Wherever  exists, denote it by f(x). Which of the following statements is true? 

Concept:

Limit of a Sequence of Functions:

1. Let  be a sequence of functions defined on a set D . We say that  converges pointwise to a function

 f  on D if, for every x  D

2. A stronger form of convergence is uniform convergence. The sequence  converges uniformly to a function  f  on D if

Explanation: The problem gives a sequence of functions defined by

and asks about the limit of  as , denoted by  . We are tasked with determining which statement about f(x) is true.

We are asked to take the limit  of the function:

As , the term . So, for large n , the function  approaches

Case 1: 

For  we have, 
 

Case 2: 
When  , the function becomes 

Therefore, as , we get f(0) = 0 .

The function f(x) ,  , is given by

This function is equal to |x| for all , 

Therefore, The correct option is 4).