I'm trying to figure out how to prove this inequality

[minifhncc]

I'm trying to figure out how to prove this inequality:

I know it's true (by graphing), but what's an algebraic way to prove it?

$$1+\frac{1}{3x^2}< x \tan \frac{1}{x} < \frac{1}{\sqrt{1-\frac{2}{3x^2}}}$$

Thanks

 

[mathman]

You need to specify the range for x. Let y=1/x, it will be easier to handle.

 

[minifhncc]

I guess it's for 'large enough x' ie. for x>N for some N.

It's actually Euler's limit that I'm trying to prove...

 

[disregardthat]

Try to differentiate the differences.