Question

URGENTEEEEEEE Pra hojee Calculo 1
Use as propriedades operatórias e as simplificações que forem necessárias para determinar os limites a seguir:
a) lim x^2−1/ x^2−3 x+2
x→1

b) lim √x−1/x−1
x→1

Answer 1

a)

$\displaystyle\lim_{x \to 1} \dfrac{x^2-1}{x^2-3x+2} \\~\\~\\ \displaystyle\lim_{x \to 1} \dfrac{(x+1)(x-1)}{x^2-x-2x+2} \\~\\~\\ \displaystyle\lim_{x \to 1} \dfrac{(x+1)(x-1)}{x(x-1)-2(x-1)}\\~\\~\\ \displaystyle\lim_{x \to 1} \dfrac{(x+1)(x-1)}{(x-1)(x-2)} \\~\\~\\ \displaystyle\lim_{x \to 1} \dfrac{(x+1)}{(x-2)} \\~\\~\\-2$

b)

$\displaystyle \lim_{x \to 1} \dfrac{\sqrt{x}-1}{x-1} \\~\\~\\\displaystyle \lim_{x \to 1} \dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{(x-1)\left(\sqrt{x}+1\right)} \\~\\~\\\displaystyle \lim_{x \to 1} \dfrac{x-1}{(x-1)\left(\sqrt{x}+1\right)}\\~\\~\\\displaystyle \lim_{x \to 1} \dfrac{1}{\sqrt{x}+1} \\~\\~\\\dfrac{1}{2}$