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## Homework Statement

Prove that :

$$\lim_{x\to 0}\frac{tan^{-1}x}{x}<1$$

And

$$\lim_{x\to 0}\frac{sin^{-1}x}{x}>1$$

## Homework Equations

None

## The Attempt at a Solution

I have to prove that arctanx has to be lesser than x.

It's derivative is 1 at x=0 and keeps decreasing as x increases. So it's slope decreases from 1.

Also, at x=0, arctanx=x. So the curve which initially touched y=x goes below it. So arctanx<x.

Is this correct?