Question

Calcule o limite a seguir:

Answer 1

Explicação passo-a-passo:

$\sf \lim_{x \to 0} \frac{sen10x}{sen7x}$

=>Valor de $\sf \lim_{x \to 0} sen10x$

$\sf \lim_{x \to 0} sen(10x) = > sen(10 \times 0) = > 0$

= >Valor de $\sf \lim_{x \to 0} sen7x$

$\sf \lim_{x \to 0} sen(7x) = > sen(7 \times 0) = 0$

Como ela é indeterminada iremos transformar a fração

$\sf \lim_{x \to 0} ( \frac{sen(10x)}{sen(7x)} \times \frac{10 \times 7x}{10 \times 7x} )$

$\sf \lim_{x \to 0} ( \frac{sen(10x) \times 10 \times 7x}{sen(7x) \times 10 \times 7x} )$

$\sf \lim_{x \to 0}( \frac{10 \times sen(10x) \times 7x}{7 \times 10x \times sen(7x)} )$

$\sf \lim_{x \to 0}( \frac{10}{7} \times \frac{sen(10x)}{10x} \times \frac{7x}{sen(7x)} )$

Agora vamos reescrever a expressão

$\sf \frac{10}{7 } \times \lim_{x \to 0}( \frac{sen(10x)}{10x} \times \frac{7x}{sen(7x)} )$

$\sf \frac{10}{7} \times \lim_{x \to 0}( \frac{sen(10x)}{10x} ) \times \lim_{x \to 0}( \frac{7x}{sen(7x)} )$

$\sf \frac{10}{7} \times \lim_{t \to 0}( \frac{sen(t)}{t} ) \times \lim_{t \to 0}( \frac{t}{sen(t)} )$

$\sf \frac{10}{7} \times 1 \times \lim_{t \to 0}(( \frac{sen(t)}{t} ) {}^{ - 1} )$

$\sf \frac{10}{7} \times 1 \times (\lim_{t \to 0}( \frac{sen(t)}{t} )) {}^{ - 1}$

$\sf \frac{10}{7} \times 1 \times {1}^{ - 1}$

$\sf \frac{10}{7}$

Limite = $\sf \frac{10}{7}$

Espero ter ajudado !!!