Question

\(\lim\limits_{x\rightarrow\sqrt{2}}\dfrac{x^2-2}{x^2-x+\sqrt{2}-2}\)

Answer 1

\(\lim\limits_{x\rightarrow\sqrt{2}}\dfrac{x^2-2}{x^2-x+\sqrt{2}-2}\)

\(=\lim\limits_{x\rightarrow\sqrt{2}}\dfrac{\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)}{\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)-\left(x-\sqrt{2}\right)}\)

\(=\lim\limits_{x\rightarrow\sqrt{2}}\dfrac{\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)}{\left(x-\sqrt{2}\right)\left(x+\sqrt{2}-1\right)}\)

\(=\lim\limits_{x\rightarrow\sqrt{2}}\dfrac{x+\sqrt{2}}{x+\sqrt{2}-1}=\dfrac{\sqrt{2}+\sqrt{2}}{\sqrt{2}+\sqrt{2}-1}=\dfrac{2\sqrt{2}}{2\sqrt{2}-1}\)