Prove or disapprove f(x) is uniformly continuous

f(x)=(sinx)/x for x not equal to zero.

and f(x)=1 for x=0

I know f(x) is uniformly continous but I can't prove it. how do i show the derivative $\displaystyle (xcosx-sinx)/x^{2}$ is bounded.

thanks

Printable View

- Mar 12th 2010, 07:41 AMcharikaarUniformly Continuous.
Prove or disapprove f(x) is uniformly continuous

f(x)=(sinx)/x for x not equal to zero.

and f(x)=1 for x=0

I know f(x) is uniformly continous but I can't prove it. how do i show the derivative $\displaystyle (xcosx-sinx)/x^{2}$ is bounded.

thanks - Mar 12th 2010, 11:33 AM1234567
- Mar 12th 2010, 11:42 AMchisigma
Take into account that the coefficients of McLaurin expansion of $\displaystyle x\cdot \cos x - \sin x$ of degree < 3 are equal to zero, so that is $\displaystyle f^{'}(0)=0$ and that both the functions $\displaystyle \frac{\cos x}{x}$ and $\displaystyle \frac{\sin x}{x^{2}}$ tend to 0 if x tends to infinity...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$