Question

calcule os limites, se exitir:

lim Sen 5x/
x->0 Sen 6x

lim x . cot x
x-0

Answer 1

Caso tenha problemas para visualizar a resposta pelo aplicativo, experimente abrir pelo navegador: https://brainly.com.br/tarefa/8089430

__________


Calcular os limites:


•   $\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{sen\,6x}}$

$=\mathsf{\underset{x\to 0}{\ell im}~sen\,5x\cdot \dfrac{1}{sen\,6x}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{5x}\cdot \dfrac{5x}{sen\,6x}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{5x}\cdot 5x\cdot \dfrac{1}{sen\,6x}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{5x}\cdot\dfrac{5\diagup\hspace{-7}x}{6\diagup\hspace{-7}x}\cdot \dfrac{6x}{sen\,6x}}$

$=\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{5x}\cdot\dfrac{5}{6}\cdot \dfrac{1}{\;\frac{sen\,6x}{6x}\;}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~\dfrac{5}{6}\cdot \dfrac{~~\frac{sen\,5x}{5x}~~}{\frac{sen\,6x}{6x}}\qquad\quad(i)}$


Calculemos separadamente os limites das funções que aparecem no numerador e no denominador:

$\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,5x}{5x}}\\\\\\ =\mathsf{\underset{u\to 0}{\ell im}~\dfrac{sen\,u}{u}=1\qquad\quad(u=5x)}$

(pelo limite trigonométrico fundamental)


e de forma análoga, também obtemos

$\mathsf{\underset{x\to 0}{\ell im}~\dfrac{sen\,6x}{6x}=1}$


Portanto, o limite $\mathsf{(i)}$ existe e é igual a

$=\mathsf{\underset{x\to 0}{\ell im}~\dfrac{5}{6}\cdot \dfrac{\underset{x\to 0}{\ell im}~\frac{sen\,5x}{5x}}{\underset{x\to 0}{\ell im}~\frac{sen\,6x}{6x}}}\\\\\\ =\mathsf{\dfrac{5}{6}\cdot\dfrac{1}{1}}\\\\\\ =\mathsf{\dfrac{5}{6}\qquad\quad\checkmark}$

________


•    $\mathsf{\underset{x\to 0}{\ell im}~x\,cot\,x}$

$=\mathsf{\underset{x\to 0}{\ell im}~x\cdot\dfrac{cos\,x}{sen\,x}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~cos\,x\cdot\dfrac{x}{sen\,x}}\\\\\\ =\mathsf{\underset{x\to 0}{\ell im}~cos\,x\cdot\dfrac{1}{\;\frac{sen\,x}{x}\;}}$

$=\mathsf{\underset{x\to 0}{\ell im}~cos\,x\cdot\dfrac{\underset{x\to 0}{\ell im}~1}{\underset{x\to 0}{\ell im}~\frac{sen\,x}{x}}}\\\\\\ =\mathsf{1\cdot \dfrac{1}{1}}\\\\\\ =\mathsf{1\qquad\quad\checkmark}$


Bons estudos! :-)


Tags:   limite função trigonométrica quociente fração mudança de variável substituição limite fundamental seno sen cosseno cos cotangente cot cotg cálculo diferencial