How do you solve the limit: lim x --> 0 sin (9x)/ sin(5x)
I really don't have much of a clue where to start, so if someone could explain every step of how to solve this, I'd be real grateful.
How do you solve the limit: lim x --> 0 sin (9x)/ sin(5x)
I really don't have much of a clue where to start, so if someone could explain every step of how to solve this, I'd be real grateful.
What I'm kind of confused on is how u factored out the 5/9 form the sin (x) equations. Could you please explain that?
Oh wait! Is the sin 9x/ 9x thing taken from the sin (x)/ x is equal to 1? But then don't you have to multiply the top and bottom both by 9x in order for it to remain equivalent? Same with the 5x?
$\displaystyle \mathop {\lim }\limits_{x \to 0} \frac{{\sin 9x}}
{{\sin 5x}} = \frac{9}
{5} \cdot \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin 9x}}
{{9x}} \cdot \frac{{5x}}
{{\sin 5x}}} \right] =$
$\displaystyle = \frac{9}
{5} \cdot \mathop {\lim }\limits_{x \to 0} \frac{{\sin 9x}}
{{9x}} \cdot \mathop {\lim }\limits_{x \to 0} \frac{{5x}}
{{\sin 5x}} =$
$\displaystyle = \frac{9}
{5} \cdot \mathop {\lim }\limits_{x \to 0} \frac{{\sin 9x}}
{{9x}} \cdot {\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin 5x}}
{{5x}}} \right)^{ - 1}} = \ldots$
Now use this
$\displaystyle {\color{red}\boxed{{\color{black}\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{ax}} = 1}}}$