Neutralization reactions:
\( \mathrm{H^+ + OH^- \rightarrow H_2O} \quad (\text{from HCl: } X), \)
\( \mathrm{2H^+ + 2OH^- \rightarrow 2H_2O} \quad (\text{from } H_2SO_4: Y). \)
From the reactions:
\( Y = 2X \implies \frac{Y}{X} = 2. \)
Final Answer: 2.
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: