무리식 극한 미정계수

[출제의도] 함수의 극한의 성질 이해하기 $\displaystyle \lim\limits _{x \rightarrow - 2} {\left( x + 2 \right)} = 0$이고 $\displaystyle \lim\limits _{x \rightarrow - 2} {\frac{x + 2}{\sqrt{x + a} - b}} = 6$ 이므로 $\displaystyle \lim\limits _{x \rightarrow - 2} {\left( \sqrt{x + a} - b \right)} = 0$ $\therefore$ $\displaystyle b = \sqrt{a - 2}$ $\displaystyle \lim\limits _{x \rightarrow - 2} {\frac{x + 2}{\sqrt{x + a} - b}}$ $\displaystyle = \lim\limits _{x \rightarrow - 2} {\frac{x + 2}{\sqrt{x + a} - \sqrt{a - 2}}}$ $\displaystyle = \lim\limits _{x \rightarrow - 2} {\frac{( x + 2 ) ( \sqrt{x + a} + \sqrt{a - 2} )}{x + 2}}$ $\displaystyle = \lim\limits _{x \rightarrow - 2} {( \sqrt{x + a} + \sqrt{a - 2} )} = 2 \sqrt{a - 2} = 6$ $\therefore$ $a = 11$, $b = 3$ 따라서 $a + b = 14$