Question

Como calcular o Limite (passo a passo por favor): lim (x-->1) (x² - 1) / (x² - x)?

Answer 1

$\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{x^{2}-1}{x^{2}-x}\\ \\ =\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{x^{2}-x+x-1}{x\cdot \left(x-1 \right )}\\ \\ =\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{x \cdot \left(x-1 \right )+1\cdot \left(x-1 \right )}{x\cdot \left(x-1 \right )}\\ \\ =\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{\left(x-1 \right )\cdot \left(x+1 \right )}{x\cdot \left(x-1 \right )}$


Cancelando o fator $\left(x-1 \right )$ no numerador e no denominador, temos

$=\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{x+1}{x}\\ \\ \right ) =\dfrac{\left(1 \right )+1}{\left(1 \right )}\\ \\ \right ) =\dfrac{2}{1}\\ \\ =2\\ \\ \\ \boxed{\underset{x \to 1}{\mathrm{\ell im}}\;\dfrac{x^{2}-1}{x^{2}-x}=2}$