1
$\begingroup$

I'm trying to find useful inequalities I can make use of when applying the squeeze theorem for multivariable limits. Is there a name for them or some good source where I can find a bunch of them? I haven't had luck googling so far.

Also, I seem to remember this one: $$ \sin x \le |x|$$ Is that one true? I've been searching online but wasn't able to find it.

$\endgroup$
0
$\begingroup$

For $|x| \geq 1$ it is clear. Else you can use the function $f(x)=x-\sin(x)$. The derivative is $1-\cos(x)$ which is non-negative, so for $x\in[0,1]$ $f\geq f(0)=0$. And on $[-1,0]$ you use the same idea with $g(x)=-x-\sin(x)$.

$\endgroup$
2
$\begingroup$

Consider the Taylor expansion:

$$\sin x = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}x^{2k+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

which is an alternating series with $x$ as an upper bound.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.