If Un = (1+1/n^2)(1+2^2/n^2)^2...(1+n^2/n^2)^n ,then

Question

If U_n = (1+\(\frac{1}{n^2}\))(1+\(\frac{2^2}{n^2}\))² ...(1+\(\frac{n^2}{n^2}\))ⁿ,then \(\lim\limits_{x \to ∞}\)\((U_n)^{\frac{-4}{n^2}}\) is equal to : 

U_n = (1+1/n²)(1+2²/n²)²...(1+n²/n²)ⁿ

(1) \(\frac{e^2}{16}\) 

(2) \(\frac{4}{e}\)

(3) \(\frac{16}{e^2}\)

(4) \(\frac{4}{e^2}\)

Answer 1

Option : (1). \(\frac{e^2}{16}\)

Put 1+x² = t

Now,

2xdx = dt