Question

Usando limites formais

Determine L= lim f(x) (com x tendendo a x0). Depois determine um número δ>0 tal que todos os valores de x
0<|x-x0|<δ ⇒ |f(x)-L|< ε

1) f(x)=3-2x, x0=3, ε=0,02

Answer 1

$\lim\limits_{x\to 3}3-2x =3-2(3)=-3$

$\forall \varepsilon \ \textgreater \ 0 ,\exists \delta \ \textgreater \ 0 : 0\ \textless \ |x-3|\ \textless \ \delta \Longrightarrow |(3-2x)-(-3)|\ \textless \ \varepsilon\\ \\ \Longrightarrow |6-2x|\ \textless \ \varepsilon\\ \Longrightarrow 2|3-x|\ \textless \ \varepsilon\\ \Longrightarrow 2|x-3|\ \textless \ \varepsilon\\ \\ \Longrightarrow |x-3|\ \textless \ \dfrac{\varepsilon}{2}\\ \\ \delta = \dfrac{\varepsilon}{2}\\ \\ \delta = \dfrac{0.02}{2}\\ \\ \boxed{\delta = 0.01}$