1. ## Calculus Proof

Prove that if there is a number B such that |f(x)| <= B for all x doesnt equal 0, then limit x->0 xf(x) = 0

Where do I start?

2. Originally Posted by Sterwine
Prove that if there is a number B such that |f(x)| <= B for all x doesnt equal 0, then limit x->0 xf(x) = 0

Where do I start?
I assume you proved that if $g(x)$ is bounded and $\lim_{x \to a}h(x) \to \infty$ or $\lim_{x \to a}h(x) \to 0$, then $\lim_{x \to a}h(x)g(x) = \lim_{x\to a}h(x)$...

If you have, simply use it and finish the proof. Otherwise, prove it and finish the proof! :P

3. Well I dont really understand how you start that but thanks for at least helping

4. Originally Posted by Sterwine
Prove that if there is a number B such that |f(x)| <= B for all x doesnt equal 0, then limit x->0 xf(x) = 0

Where do I start?
since $\lim_{x\to0}x=0$ then for each $\epsilon>0$ we can let $|x|<\frac\epsilon{B}.$

on the other hand, all we need to prove is that $|xf(x)|<\epsilon,$ in efect $\left| xf(x) \right|=\left| x \right|\left| f(x) \right|<\frac{\epsilon }{B}\cdot B=\epsilon,$ and we're done.