A question on limits using the epsilon-delta definition

Question

I am trying to solve this problem but I keep getting stuck. This question does not seem to be particularly hard.
The question is stated below:
$ |f(x)| < \frac{1}{100} $ if $0 < x < \delta$
Find $\delta$ if $f(x) = x\sin(x)$

Please do not tell me the answer directly but help me understand it and point me in the right direction. That way I will learn more.

Thank you for helping and stay safe!

Answer 1

Hint: Think of what happens as you increase $x$. Up until what value of $x$ can you safely say that $|x \sin(x)|$ will always be less than 0.01?

Answer 2

You may notice that $|\sin x|\le1$ and so $|f(x)|=|x\sin x|=|x||\sin x|\le|x|$.

Since $x>0,x=|x|$ and we have $|x|<\delta$. Thus $|f(x)|<\delta$ whenever $0<x<\delta$.

Can we select an appropriate positive $\delta$ to make $|f(x)|<1/100$ true?

Answer 3

Hint:

$|\sin x|\le |x|$ for all $x$, so $|f(x)|\le\cdots$