For each n ≥ 1 define fn : ℝ → ℝ by \(\rm f_n(x)=\frac{x^2}{√{x^2+\frac{1}{n}}}, \) x ∈ ℝ
where √ denotes the non-negative square root. Wherever \(\rm \lim_{n \rightarrow \infty}f_n(x)\) exists, denote it by f(x). Which of the following statements is true?
1
There exists x ∈ ℝ such that f(x) is not defined
2
f(x) = 0 for all x ∈ ℝ
3
f(x) = x for all x ∈ ℝ
4
f(x) = |x| for all x ∈ ℝ