Question

Solve lim x=0 sin6x/sin7x

Answer 1

EXPLANATION.

$\sf \implies \lim_{x \to 0} Sin(6x)/Sin(7x).$

Put x = 0 in equation we get,

$\sf \implies \lim_{x \to 0} \dfrac{Sin6(0)}{Sin7(0)} .$

As we can see it is in the Form of 0/0 form,

According to 0/0 form,

Simply Factorizes the equation, we get.

Multiply and divide the numerator and denominator by 7x, we get.

$\sf \implies \lim_{x \to 0}\dfrac{Sin(6x).(7x)}{Sin(7x).(7x)} .$

Again, we can multiply and divide the numerator and denominator by 6x, we get.

$\sf \implies \lim_{x \to 0} \dfrac{Sin(6x).(7x).(6x)}{Sin(7x).(7x).(6x)}.$

Separates each equation, we get.

$\sf \implies \lim_{x \to 0} \dfrac{Sin(6x)}{6}. \ \lim_{x \to 0} \dfrac{7x}{Sin(7x)} .\ \lim_{x \to 0} \dfrac{6x}{7x}$

$\sf As \ we \ know\ that = \lim_{x \to 0} \dfrac{Sin(x)}{x} = 1 .$

$\sf As\ we\ know\ that = \lim_{x \to 0} \dfrac{x}{Sin(x)} = 1.$

By Applying this formula in equation, we get.

$\sf \implies 1.1. \lim_{x \to 0} \dfrac{6x}{7x} .$

As we know that 6/7 is the constant term,

Then equation is written As,

$\sf \implies\dfrac{6}{7} \lim_{x \to 0} \dfrac{x}{x}.$

$\sf value \ of \ equation =\lim_{x \to 0} \dfrac{Sin(6x)}{Sin(7x)} = \dfrac{6}{7} .$