Question

limite de 1/(x-1) + (x-5)/(x^2+2x-3) imagem em anexo.

Answer 1

Realizando algumas operações


$\displaystyle\lim_{x\to1}\left ( \frac{1}{x-1}+\frac{x-5}{x^2+2x-3} \right)\\\\\\\lim_{x\to1}\left ( \frac{1}{x-1}+\frac{x-5}{(x+3)(x-1)} \right)\\\\\\\lim_{x\to1}\left ( \frac{x+3}{(x+3)(x-1)}+\frac{x-5}{(x+3)(x-1)} \right)\\\\\\\lim_{x\to1}\left(\frac{2x-2}{(x+3)(x-1)}\right)\\\\\\\lim_{x\to1}\left(\frac{2(x-1)}{(x+3)(x-1)}\right)$


Quando x tende a 1, temos x ≠ 1, logo x - 1 ≠ 0. Portanto podemos cancelar o fator comum


$\displaystyle\lim_{x\to1}\left(\frac{2(x-1)}{(x+3)(x-1)}\right)=\lim_{x\to1}\left(\frac{2}{x+3}\right)=\frac{1}{2}$