Limit of sin(x)cos(x)/x as x approaches 0?

[jog511]

Homework Statement

lim x->0 sinxcosx/x

Homework Equations

lim x->0 sinx/x = 1

The Attempt at a Solution

Pretty sure I need to use above property but I believe cosx/x is undef.

 

[pasmith]

Homework Statement

lim x->0 sinxcosx/x

Homework Equations

lim x->0 sinx/x = 1

The Attempt at a Solution

Pretty sure I need to use above property but I believe cosx/x is undef.

Recall the identity $\sin(2x) = 2\sin x \cos x$.

 

[LCKurtz]

Or write it as$$
\frac {\sin x} x \cdot \cos x$$

 

[jog511]

limx->o sinx/x * limx->0 cosx = 1*1 =1

 

[dirk_mec1]

Correct.

 

[HallsofIvy]

That's the simplest way to do it but using pasmith's suggestion, since sin(2x)= 2sin(x)cos(x), sin(x)cos(x)= sin(2x)/x so that sin(x)cos(x)/x= sin(2x)/2x. Now let u= 2x. As x goes to 0, so does u= 2x and we have
$$\lim_{x\to 0} \frac{sin(x)cos(x)}{x}= \lim_{u\to 0}\frac{sin(u)}{u}= 1$$.