1. ## proving uniformly continousity

i need to prove that $\displaystyle f(x)=ln(x^{2}+cos^{2}x)$ is uniformly continous. now, i thought of a way. i know that $\displaystyle f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}$
so i acctually need to prove that $\displaystyle f'(x)$ is bounded ($\displaystyle N\leq f'(x)\leq M$). but how do i prove it?

2. ## Re: proving uniformly continousity

Originally Posted by orir
i need to prove that $\displaystyle f(x)=ln(x^{2}+cos^{2}x)$ is uniformly continous. now, i thought of a way. i know that $\displaystyle f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}$
so i acctually need to prove that $\displaystyle f'(x)$ is bounded ($\displaystyle N\leq f'(x)\leq M$). but how do i prove it?
Have a look at the graph of the derivative.

3. ## Re: proving uniformly continousity

i took a look, and i can see that its limit in infinity (and -infinity) is 0 (is that what you're implying of?) . but as i posted in another post, i don't really know how to solve this limit by myself.
is there any other way except the one with the limit?

4. ## Re: proving uniformly continousity

Originally Posted by orir
i took a look, and i can see that its limit in infinity (and -infinity) is 0 (is that what you're implying of?) . but as i posted in another post, i don't really know how to solve this limit by myself.
is there any other way except the one with the limit?
I was simply pointing out that the derivative is in fact bounded.

That derivative is continuous. and the limit is $\displaystyle 0$ as $\displaystyle x\to \pm\infty.$

5. ## Re: proving uniformly continousity

but why is that so?
and, is it enough to say it's bounded?

6. ## Re: proving uniformly continousity

Originally Posted by orir
i need to prove that $\displaystyle f(x)=ln(x^{2}+cos^{2}x)$ is uniformly continous. now, i thought of a way. i know that $\displaystyle f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}$
so i acctually need to prove that $\displaystyle f'(x)$ is bounded ($\displaystyle N\leq f'(x)\leq M$). but how do i prove it?
See the following very similar post:
Uniform Continuity of $\sqrt{|x|}$ on $\mathbb{R}$
and see "The more sophisticated way" part in
Uniform Continuity of $\sqrt{|x|}$ on $\mathbb{R}$