무리식의 극한

$\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{\sqrt{x + \alpha ^{2}} - \sqrt{x + \beta ^{2}}}{\sqrt{4 x + \alpha} - \sqrt{4 x + \beta}}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left( \frac{\sqrt{x + \alpha ^{2}} - \sqrt{x + \beta ^{2}}}{\sqrt{4 x + \alpha} - \sqrt{4 x + \beta}} \times \frac{\sqrt{x + \alpha ^{2}} + \sqrt{x + \beta ^{2}}}{\sqrt{4 x + \alpha} + \sqrt{4 x + \beta}} \times \right.$ $\displaystyle \left. \frac{\sqrt{4 x + \alpha} + \sqrt{4 x + \beta}}{\sqrt{x + \alpha ^{2}} + \sqrt{x + \beta ^{2}}} \right)$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{( \alpha ^{2} - \beta ^{2} ) ( \sqrt{4 x + \alpha} + \sqrt{4 x + \beta} )}{( \alpha - \beta ) ( \sqrt{x + \alpha ^{2}} + \sqrt{x + \beta ^{2}} )}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left\{ ( \alpha + \beta ) \frac{\left( \sqrt{4 + \frac{\alpha}{x}} + \sqrt{4 + \frac{\beta}{x}} \right)}{\left( \sqrt{1 + \frac{\alpha ^{2}}{x}} + \sqrt{1 + \frac{\beta ^{2}}{x}} \right)} \right\}$ $\displaystyle = \frac{1 \cdot ( 2 + 2 )}{1 + 1} = 2$