Question

Demonstre cada afirmação usando a definição ε, δ de limite.
a) $\lim_{x \to \1} \frac{2 + 4x}{3}= 2$

Answer 1

    $\lim\limits_{x\to1}\dfrac{2+4x}{3}=2\iff (\forall \varepsilon \ \textgreater \ 0)(\exists \delta \ \textgreater \ 0):|x-1|\ \textless \ \delta\Rightarrow \left|\dfrac{2+4x}{3}-2\right|\ \textless \ \varepsilon\\ \\ \\ \\ \left|\dfrac{2+4x}{3}-2\right|\ \textless \ \varepsilon\\ \\ \\ |4x-4|\ \textless \ 3\varepsilon\\ \\ |x-1|\ \textless \ \dfrac{3}{4}\varepsilon\\ \\ \\ \texttt{Entonces existe un }\delta =\dfrac{3}{4}\varepsilon$