Question

Por favor respondam pra mim a questão do arquivo

Answer 1

1 - A função é contínua à direita e à esquerda de 1.

$f(1)=x^{2} \\ \\ f(1)=1 \\ \\ ----- \\ \\ \displaystyle \lim_{x \to 1^{-}} -x^{3}+2=1 \\ \\ \displaystyle \lim_{x \to 1^{-}} f(x)=f(1) \therefore \displaystyle \lim_{x \to 1^{-}} f(x) = \exists \\ \\ ----- \\ \\ \displaystyle \lim_{x \to 1^{+}} x^{2}=1 \\ \\ \displaystyle \lim_{x \to 1^{+}} f(x)=f(1) \therefore \displaystyle \lim_{x \to 1^{+}} f(x)= \exists \\ \\ ----- \\ \\ \displaystyle \lim_{x \to 1^{-}}f(x)=\displaystyle \lim_{x \to 1^{+}}f(x) \therefore \displaystyle \lim_{x \to 1} f(x)=1$


2 - 

a)

$\displaystyle \lim_{x \to 0^{+}} \sqrt{2x} =0$

b)

$\displaystyle \lim_{x \to 5^{-}} \frac{|x-5|}{5-x} \\ \\ \displaystyle \lim_{x \to 5^{-}} \frac{5-x}{5-x} =1 \\ \\ ------ \\ \\ \displaystyle \lim_{x \to 5^{+}} \frac{|x-5|}{5-x} \\ \\ \displaystyle \lim_{x \to 5^{+}} \frac{-|5-x|}{5-x} \\ \\ \displaystyle \lim_{x \to 5^{+}} -\frac{5-x}{5-x}=-1 \\ \\ ------ \\ \\ \displaystyle \lim_{x \to 5^{-}} f(x) \neq \displaystyle \lim_{x \to 5^{+}} f(x) \therefore \displaystyle \lim_{x \to 5} f(x) = \nexists$