N[Limit[((Abs[(x^2)×(Sin[10Log[x]]+1)] + 1000000000x)/(Abs[(x^2)×(Cos[10Log[x]]+1)] + 1000000000x)), x -> ∞]]

Plot[{Abs[(x^2)×(Sin[10Log[x]]+1)] + 1000000000x,Abs[(x^2)×(Cos[10Log[x]]+1)] + 1000000000x},{x, 1, 1000000000}]

Plot[Tooltip[{Abs[(x^2)×(Sin[10Log[x]]+1)] + 1000000000x,1000000000x,2 (x^1.96)+1000000000 x,2 (x^2)+1000000000 x,Abs[(x^2)×(Cos[10Log[x]]+1)] + 1000000000x}],{x, 1, 1000000000}]

Limit[(Abs[(x^2)×(Sin[10Log[x]]+1)] + 1000000000x)/(1000000000x),{x -> ∞}]

Should be: {Interval[{1,∞}]}

Limit[(Abs[(x^2)×(Sin[10Log[x]]+1)] + 1000000000x)/(2 x^2+1000000000x),{x -> ∞}]