Show this equality of limits - cross products

[Ted123]

Homework Statement

Show that

$\displaystyle\lim_{\underline{h}\to 0} \frac{\underline{c} \times \underline{h}}{\|\underline{h}\|} =0$

where c and h are vectors and x denotes cross product.

Homework Equations

The Attempt at a Solution

No idea how to do this?

 

[jbunniii]

Homework Statement

Show that

$\displaystyle\lim_{\underline{h}\to 0} \frac{\underline{c} \times \underline{h}}{\|\underline{h}\|} =0$

where c and h are vectors and x denotes cross product.

Homework Equations

The Attempt at a Solution

No idea how to do this?

I don't think it's true. Suppose the vectors are in R^3, with $i,j,k$ the usual coordinate vectors. Let $c = i$, $h = tj$. (where $t$ is a scalar)

Then

$$\left|\frac{c \times h}{|h|}\right| = \left|\frac{tk}{|tj|}\right| = \frac{|tk|}{|tj|} = \frac{|t|}{|t|} = 1$$

so the limit is certainly not zero as $h$ approaches zero along the $i$ coordinate axis. If the limit were zero, it would have to be zero no matter how $h$ approaches zero.

 

[LCKurtz]

What happens if you let

$$\vec c = \vec i,\, \vec h = \lambda \vec j$$

where $\vec i,\vec j$ are the standard unit vectors?