# Theorem for limits?

1. Feb 13, 2015

### Mr Davis 97

I read in a calculus book that. "Given $\lim_{x \to a}\frac{f(x)}{g(x)} = c(c\neq 0)$, when $\lim_{x \to a}g(x) = 0$, then $\lim_{x \to a}f(x) = 0$. Why is this true?

2. Feb 13, 2015

### DEvens

3. Feb 13, 2015

### Staff: Mentor

A non-rigorous explanation is that, since $\frac{f(x)}{g(x)} \to c$, where c ≠ 0, then f and g are approximately equal near a. If g approaches zero as x approaches a, then so does f.

4. Feb 13, 2015

### PeroK

Have you tried finding a counterexample? Usually a good way to see why something is true is to try to show that it's false.

And, why must you have $c \ne 0$?

5. Feb 13, 2015

### Staff: Mentor

Last edited by a moderator: May 7, 2017
6. Feb 13, 2015

### mathman

Multiply by g(x). Limit for f(x) = c(limit for g(x)) = 0.