# Thread: Prove a limit of 2 functions multiplied is 0

1. ## Prove a limit of 2 functions multiplied is 0

R= Reals
Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
x --> c g=0. Prove that the limit as x --> c fg=0

2. Originally Posted by gixxer998
R= Reals
Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
x --> c g=0. Prove that the limit as x --> c fg=0
If $f$ is bounded on a neighborhood of $c$, then $-M\leq f(x)\leq M$ for some $M\in\mathbb{R}$. So $\lim_{x\to c}-Mg(x)\leq \lim_{x\to c}f(x)g(x)\leq \lim_{x\to c}Mg(x)$

Proceed using properties of limits.

3. Originally Posted by gixxer998
R= Reals
Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
x --> c g=0. Prove that the limit as x --> c fg=0
Hint: $|fg(x)-0|=|f(x)g(x)|\le B|g(x)|$