Question

Calcule o limite: lim x --> a cosx - cos a / x - a

Answer 1

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$\displaystyle\sf{\lim_{x \to a}\dfrac{cos(x)-cos(a)}{x-a}}$

$\sf{fac_{\!\!,}a~t=x-a\implies x=t+a}\\\sf{x\to a~quando~t\to 0}$

$\displaystyle\sf{\lim_{x \to a}\dfrac{cos(x)-cos (a)}{x-a}=\lim_{t \to 0}\dfrac{cos(t+a)-cos(a)}{t}}$

$\displaystyle\sf{\lim_{t \to 0}\dfrac{cos(t)\cdot cos(a)-sen(t)\cdot sen(a)-cos(a)}{t}}\\\displaystyle\sf{\lim_{t \to 0}\dfrac{cos(t)\cdot cos (a)-cos(a)- sen(t)\cdot sen(a)}{t}}$

$\displaystyle\sf{\lim_{t \to 0}cos(a)\cdot(-1)\lim_{t \to 0}\dfrac{cos(a)[1-cos(t)]}{t}-\lim_{t \to 0}\dfrac{sen(t)}{t}\cdot\lim_{t \to 0} cos(a)}\\\sf{cos(a)\cdot(-1)\cdot0-1\cdot cos(a)=-cos(a)\checkmark}$