Question

lim tb x - cos x / x^2 sendo que x tende a 0

Answer 1

$\mathsf{ \lim_{x \to 0}~ tg(x)-\dfrac{cos(x)}{x^2}}\\\\\mathsf{ \lim_{x \to 0}~ tg(x)-\lim_{x \to 0}\dfrac{cos(x)}{x^2} }\\\\\\\mathsf{\lim_{x \to 0}~tg(x)=0}\\\\\mathsf{\lim_{x \to 0}\dfrac{cos(0)=1}{0^2}=\infty}\\\downarrow\\\mathsf{Pela~teorema~dos~limites,~quando~um~inteiro~\'e~dividido~por~um~n\'umero}\\\mathsf{muito~pr\'oximo~de~0,~resultara~um~n\'umero~infinito!}\\\\\\\boxed{\mathsf{\lim_{x\to0^+} 0-\infty=-\infty}}$


$\mathsf{Teorema:}\\\\\\\mathsf{\lim_{x\to0^\pm}\dfrac{n}{x^2}=\infty~\gets~Se~n~\'e~um~inteiro~positivo~e~x~tende~a~0~tanto~pela}\\\mathsf{~~~~~~~~~~~~~~~~~~~~~~~~~~~~direita~quanto~pela~esquerda,~resultara~um~n\'umero}\\\mathsf{~~~~ ~~~~~~~ ~~~~~~ ~~~~ ~~~~~~~infinito~positivo}$

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