55. lim(x→1) (√(x² + 3) - x - 1)/(1 - x²) = .....

Explanation:

Step1: 分母を因数分解

$1 - x^{2}=(1 - x)(1 + x)$

Step2: 分子に共役式をかける

分子$\sqrt{x^{2}+3}-x - 1$に$\sqrt{x^{2}+3}+x + 1$をかける。 [ \begin{align*} &(\sqrt{x^{2}+3}-x - 1)(\sqrt{x^{2}+3}+x + 1)\ =&(x^{2}+3)-(x + 1)^{2}\ =&x^{2}+3-(x^{2}+2x + 1)\ =&x^{2}+3 - x^{2}-2x - 1\ =&2-2x \end{align*} ]

Step3: 極限を求める

[ \begin{align*} &\lim_{x\rightarrow1}\frac{\sqrt{x^{2}+3}-x - 1}{1 - x^{2}}\ =&\lim_{x\rightarrow1}\frac{2-2x}{(1 - x)(1 + x)(\sqrt{x^{2}+3}+x + 1)}\ =&\lim_{x\rightarrow1}\frac{2(1 - x)}{(1 - x)(1 + x)(\sqrt{x^{2}+3}+x + 1)}\ =&\lim_{x\rightarrow1}\frac{2}{(1 + x)(\sqrt{x^{2}+3}+x + 1)}\ =&\frac{2}{(1 + 1)(\sqrt{1^{2}+3}+1 + 1)}\ =&\frac{2}{2(\sqrt{4}+2)}\ =&\frac{2}{2(2 + 2)}\ =&\frac{2}{8}\ =&\frac{1}{4} \end{align*} ]

Answer:

$\frac{1}{4}$