What is $lim_(x->0) sin(sin3x)/(sin 7x)$ ?

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Answer 1 · 2017-09-22T21:54:06

$3/7$

My first impression is that:

$lim_(x->0) sin(sin3x)/(sin 7x) = 3/7$

since $sin theta$ behaves like $theta$ for small values of $theta$ .

Let us start by assuming:

$lim_(t->0) sin t / t = 1$

Then:

$lim_(x->0) sin(sin3x)/(sin7x)$

$= lim_(x->0) (sin(sin3x)/(sin3x) * (sin3x)/(3x) * (7x)/(sin 7x)) * 3/7$

$= 1 * 1 * 1 * 3/7$

$= 3/7$

What is lim_(x->0) sin(sin3x)/(sin 7x)lim_(x->0) sin(sin3x)/(sin 7x) ?