# Prove limit x approaches 0 of a rational function = ratio of derivatives

1. Oct 23, 2014

### FlorenceC

1. The problem statement, all variables and given/known dat
If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that

lim f(x) = f'(0)
x->0 g(x) g'(0)

3. The attempt at a solution
I know that lim as x→a f(a) = f(a) if function is continuous. since its differentiable it's continuous. so lim x→0 f(x) = f(0). and lim x→0 g(x) = g(0) but you can't have a 0/0.
I have so idea how to get to the derivative part.

2. Oct 23, 2014

### PeroK

This is essentially a special case of L'Hospital's rule. Do you know the Mean Value Theorem?

3. Oct 23, 2014

### FlorenceC

Yes. but how does mvt apply? we didn't learn lhopitals rule yet.

4. Oct 23, 2014

### PeroK

Actually, you don't even need the MVT. Just think about the definition of a derivative.

5. Oct 23, 2014

### RUber

This is one of those cases where adding zero back into the expression makes it more clear:
$\lim_{a\to 0} f(a) = \lim_{a\to 0} f(0+a)-f(0)$ ...