무리식 극한 미정계수

$x \rightarrow 1$일 때 (분모) → 0 이므로 (분자) → 0 이어야 한다. 따라서 $\displaystyle \sqrt{1 + a} - b = 0 , b = \sqrt{1 + a}$ $\displaystyle \lim\limits _{x \rightarrow 1} \frac{\sqrt{x ^{2} + a} - b}{x - 1}$ $\displaystyle = \lim\limits _{x \rightarrow 1} \frac{\sqrt{x ^{2} + a} - \sqrt{1 + a}}{x - 1}$ $\displaystyle = \lim\limits _{x \rightarrow 1} \frac{x ^{2} - 1}{( x - 1 ) \sqrt{x ^{2} + a} + \sqrt{1 + a}}$ $\displaystyle = \lim\limits _{x \rightarrow 1} \frac{x + 1}{\sqrt{x ^{2} + a} + \sqrt{1 + a}}$ $\displaystyle = \frac{2}{\sqrt{1 + a} + \sqrt{1 + a}}$ $\displaystyle = \frac{1}{\sqrt{1 + a}} = \frac{1}{2}$ $\therefore 1 + a = 4 , a = 3 , b = 2$ $\therefore ab = 6$