Question

lim x tende para pi de f(x) +2÷ x-pi
f(x) = -2 +4 sen x​

Answer 1

$\displaystyle\mathsf{\lim_{x \to \pi}\dfrac{f(x)+2}{x-\pi}}$

$\displaystyle\mathsf{\lim_{x \to \pi}\dfrac{-2+4sen(x)+2}{x-\pi}}$

$\displaystyle\mathsf{\lim_{x \to \pi}\dfrac{4sen(x) }{x-\pi}}$

Fazendo

$\mathsf{u=x-\pi}\\\mathsf{x=u+\pi~x\to~\pi~quando~u\to\,0}$

$\displaystyle\mathsf{\lim_{x \to \pi}\dfrac{4sen(x) }{x-\pi}}\\=\displaystyle\mathsf{\lim_{u \to 0}\dfrac{4sen(u+\pi) }{u}}$

$\displaystyle\mathsf{4\lim_{u \to 0}\dfrac{sen(u+\pi) }{u}} \\\displaystyle\mathsf{4\lim_{u \to 0}\dfrac{sen(u).cos(\pi) + sen(\pi).cos(u) }{u}}$

$\displaystyle\mathsf{4\lim_{u \to 0}\dfrac{ - sen(u)}{u}} \\\displaystyle\mathsf{ - 4\lim_{u \to 0}\dfrac{ sen(u)}{u} = - 4.1 = - 4}$