1. Lt n→ ∞ sin 2[π√((n!)2-n!)]2. Ltx→0 ({x})1/x + (1/x){x} , (x>0,{.} denotes fractional part of x)please solve both the Questions.

Dear sindhu

first question

Lt _{n→ ∞} sin ²  [π√((n!)²-n! )]

or Lt _{n→ ∞} sin ²  [∏n!√( 1-1/n! )]

apply limit part inside the root will become 1

and n! is very large nimber but it is a even number.

so Lt _{n→ ∞} sin ²  [∏n!√( 1-1/n! )] = sin ²  [∏*even number]

                                                        =0

For second question

Lt_x→0 ({x})^1/x + (1/x)^{x}

as limit tends to zero {x}=x

  so Lt_x→0 ({x})^1/x + (1/x)^{x} =Lt_x→0 (x)^1/x + (1/x)^x

   apply limit

      Lt_x→0 (x)^1/x + (1/x)^x       = 0 +1=1

Approved