Question

Determine o valor de k, pertencente aos números reais, para que a função
abaixo seja contínua.
f(x){ sen(x) +cos (x); se x ≤0
{k-sen^2 (x), se x >0

Answer 1

Função contínua :

$\displaystyle \lim_{\text x\to 0^{+ }}\text{f(x)} = \lim_{\text x\to 0^{-}}\text{f(x)}=\text{f(0)}$

$\displaystyle \text{f(x)}=\left \{ {{\text{sen(x) + cos(x)} \ , \ \ \text{se x }\leq 0} \atop {\text k-\text{sen}^2(\text x)\ , \ \ \text{se x }>0}} \right. \\\\\\ {\text k = \ ? } \\\\\\ \lim_{\text x\to 0}\ \text{sen(x) + cos(x) } = \lim_{\text x\to 0} \ \text k -\text{sen}^2(\text x) \\\\ \text{sen(0)+cos(0)} =\text k -\text{sen}^2(0) \\\\ 0+1=\text k-0 \\\\ \huge\boxed{\text k = 1}\checkmark$