무리식 극한의 수렴조건으로 미지수 결정

조건 (가)에서 $\displaystyle \lim\limits _{x \rightarrow \infty} {} \left( \sqrt{\left( a - b \right) x ^{2} + ax} - x \right)$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \frac{\left\{ \sqrt{\left( a - b \right) x ^{2} + ax} - x \right\} \left\{ \sqrt{\left( a - b \right) x ^{2} + ax} + x \right\}}{\sqrt{\left( a - b \right) x ^{2} + ax} + x}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \frac{\left( a - b - 1 \right) x ^{2} + ax}{\sqrt{\left( a - b \right) x ^{2} + ax} + x}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \frac{\left( a - b - 1 \right) x + a}{\sqrt{a - b + \frac{a}{x}} + 1}$ 이므로 $\displaystyle \lim\limits _{x \rightarrow \infty} {} \frac{\left( a - b - 1 \right) x + a}{\sqrt{a - b + \frac{a}{x}} + 1}$가 수렴하려면 $a - b - 1 = 0$ $\therefore$ $a - b = 1$ $\cdots$$\cdots$ ㉠ $\displaystyle \lim\limits _{x \rightarrow \infty} {} \frac{\left( a - b - 1 \right) x + a}{\sqrt{a - b + \frac{a}{x}} + 1}$$\displaystyle = \lim\limits _{x \rightarrow \infty} {} \frac{a}{\sqrt{1 + \frac{a}{x}} + 1}$$\displaystyle = \frac{a}{\sqrt{1} + 1}$ 이므로 $\displaystyle \frac{a}{2} = c$에서 $a = 2 c$, $b = 2 c - 1$ $\cdots$$\cdots$ ㉡ 조건 (나)에서 $x = - t$라 하면 $x \rightarrow - \infty$일 때 $t \rightarrow \infty$이므로 $\displaystyle \lim\limits _{x \rightarrow - \infty} {} \left( ax - b - \sqrt{- \left( b + 1 \right) x ^{2} - 4 x} \right)$ $\displaystyle = \lim\limits _{t \rightarrow \infty} {} \left\{ - at - b - \sqrt{- \left( b + 1 \right) t ^{2} + 4 t} \right\}$ $\displaystyle = \lim\limits _{t \rightarrow \infty} {} \frac{\left\{ at + b + \sqrt{- \left( b + 1 \right) t ^{2} + 4 t} \right\} \left\{ at + b - \sqrt{- \left( b + 1 \right) t ^{2} + 4 t} \right\}}{- at - b + \sqrt{- \left( b + 1 \right) t ^{2} + 4 t}}$ $\displaystyle = \lim\limits _{t \rightarrow \infty} {} \frac{\left( at + b \right) ^{2} - \left\{ - \left( b + 1 \right) t ^{2} + 4 t \right\}}{- at - b + \sqrt{- \left( b + 1 \right) t ^{2} + 4 t}}$ $\displaystyle = \lim\limits _{t \rightarrow \infty} {} \frac{\left( a ^{2} + b + 1 \right) t + 2 \left( ab - 2 \right) + \frac{b ^{2}}{t}}{- a - \frac{b}{t} + \sqrt{- \left( b + 1 \right) + \frac{4}{t}}}$ 이므로 $\displaystyle \lim\limits _{t \rightarrow \infty} {} \frac{\left( a ^{2} + b + 1 \right) t + 2 \left( ab - 2 \right) + \frac{b ^{2}}{t}}{- a - \frac{b}{t} + \sqrt{- \left( b + 1 \right) + \frac{4}{t}}}$가 수렴하려면 $a ^{2} + b + 1 = 0$ $\cdots$$\cdots$ ㉢ ㉠, ㉢을 연립하여 풀면 $a = 0$, $b = - 1$ 또는 $a = - 1$, $b = - 2$ 이때 $a = 0$, $b = - 1$이면 $\displaystyle \lim\limits _{t \rightarrow \infty} {} \frac{\left( a ^{2} + b + 1 \right) t + 2 \left( ab - 2 \right) + \frac{b ^{2}}{t}}{- a - \frac{b}{t} + \sqrt{- \left( b + 1 \right) + \frac{4}{t}}}$$= - \infty$ 이므로 조건을 만족시키지 않는다. $\therefore$ $a = - 1$, $b = - 2$, $\displaystyle c = - \frac{1}{2}$ ($\because$ ㉡) 즉 $\displaystyle \lim\limits _{t \rightarrow \infty} {} \frac{\left( a ^{2} + b + 1 \right) t + 2 \left( ab - 2 \right) + \frac{b ^{2}}{t}}{- a - \frac{b}{t} + \sqrt{- \left( b + 1 \right) + \frac{4}{t}}}$$\displaystyle = \frac{2 \left( ab - 2 \right)}{- a + \sqrt{- \left( b + 1 \right)}}$$= d$ 에서 $d = 0$ $\therefore$ $a + b + c + d$$\displaystyle = \left( - 1 \right) + \left( - 2 \right) + \left( - \frac{1}{2} \right) + 0$$\displaystyle = - \frac{7}{2}$