The balanced equation says that 2 moles C₂H₆ and 7 moles O₂ react together, i.e. in a ratio of 7:2 or 3.5 moles of O₂ to C₂H₆.
With molar masses 30.07 g/mol (C₂H₆) and 31.998 g/mol (O₂), the given quantities amount to
(19 g C₂H₆) × (1/30.07 mol/g) ≈ 0.63 mol C₂H₆
(115 g O₂) × (1/31.998 mol/g) ≈ 3.59 mol O₂
Now, 0.63/2 ≈ 0.32, and for every 0.32 mol C₂H₆ consumed, the reaction requires 7×0.32 ≈ 2.2 mol O₂. Then in order to consume all of the C₂H₆, the reaction would need 2×2.2 ≈ 4.4 mol O₂, which we don't have.
In other words, we have too much C₂H₆ and not enough O₂, so O₂ is the limiting reactant.
