Calcula el següent límit:
$$\displaystyle\lim_{x \to 3}{\dfrac{x-3}{1-\sqrt{x-2}}}$$
Desenvolupament:
$$\displaystyle\lim_{x \to 3}{\dfrac{x-3}{1-\sqrt{x-2}}}=\dfrac{0}{0}$$
Multipliquem i dividim pel conjugat:
$$\displaystyle\lim_{x \to 3}{\dfrac{(x-3)(1+\sqrt{x-2})}{(1-\sqrt{x-2})(1+\sqrt{x-2})}}=\lim_{x \to 3}{\dfrac{(x-3)(1+\sqrt{x-2})}{1^2-(\sqrt{x-2})^2}}=$$
$$=\displaystyle\lim_{x \to 3}{\dfrac{(x-3)(1+\sqrt{x-2})}{1-x+2}}=\lim_{x \to 3}{\dfrac{(x-3)(1+\sqrt{x-2})}{-(x-3)}}=$$
$$=\displaystyle\lim_{x \to 3}{-(1+\sqrt{x-2})}=-(1+\sqrt{3-2})=-(1+1)=-2$$
Solució:
$$-2$$