1^ste manier :
\(\displaystyle\lim_{x\to1}\frac{\sqrt x-1}{x-1}\;+\;\displaystyle \lim_{x\to 1}\frac{x^2-1}{x-1}=\displaystyle\lim_{x \to 1}\frac{(\sqrt x-1)(\sqrt x+1)}{(x-1)(\sqrt x+1)}+\displaystyle \lim_{x \to 1}\frac{(x^2-1)(x+1)}{(x-1)(x+1)}\\=\displaystyle \lim_{x \to 1}\frac{1}{\sqrt x+1}+\displaystyle \lim_{x \to 1}\frac{x+1}{1}=\frac12+2=\frac52\)
2^de manier :
\(\displaystyle \lim_{x \to 1}\frac{\sqrt x-1}{x-1}\;+\;\displaystyle \lim_{x \to 1}\frac{x^2-1}{x-1}\overset{H}{=}\displaystyle \lim_{x\to1}\frac{\frac1{2\sqrt x}}{1}+\displaystyle \lim_{x \to 1}\frac{2x}{1}=\frac12+2=\frac52\)
3^de manier :
\(\displaystyle\lim_{x\to1}\frac{\sqrt x-1}{x-1}\;+\;\displaystyle\lim_{x\to1}\frac{x^2-1}{x-1}=\displaystyle\lim_{x\to1}\frac{\sqrt x+x^2-2}{x-1}=\displaystyle \lim_{x\to 1}\frac{x^2-1+\sqrt x-1}{x-1}\\=\displaystyle \lim_{x\to 1}\frac{(x-1)(x+1)+(\sqrt x-1)}{x-1}=\displaystyle\lim_{x\to 1}\frac{(\sqrt x-1)(\sqrt x+1)(x+1)+(\sqrt x-1)}{(\sqrt x-1)(\sqrt x+1)}\\=\displaystyle\lim_{x\to1}\frac{(\sqrt x+1)(x+1)+1}{\sqrt x+1}=\frac{2.2+1}{2}=\frac52\)