Limit of quotient of two functions

1. May 24, 2005

ModernLove

Let f(x) and g(x) be functions.

Then if limit of f(x)/g(x) = 1. That implies lim f(x) = lim g(x) right?

Consider this proof.

lim f(x)/g(x) = 1
lim f(x) x lim 1/g(x) = 1
lim f(x) = 1 / (lim 1/g(x))
lim f(x) = lim g(x).

2. May 24, 2005

arildno

No, the implication doesn't follow since the limits of f and g might not exist in the first place at the point where the limit of the quotient is 1.

3. May 24, 2005

uart

Provided that the indivual limits actually exist then yes they will be equal. But just because the limit of f/g exists it doesn't mean that the limits of f and g neccessarily exist.

EDIT : No I'm not turning into a parrot, I must have posted the same time as arildno. :)

Last edited: May 24, 2005
4. May 24, 2005

arildno

5. May 24, 2005

uart

Or in this case just stating the obvious I think. :)

6. May 24, 2005

whozum

$$\lim_{x\rightarrow 1} \frac{\sin x}{x} = 1$$

$$\lim_{x\rightarrow 1} \sin x \neq 1$$

Can a mathematician clarify?

7. May 24, 2005

arildno

Eeh, you've got:
$$\lim_{x\to0}\frac{\sin(x)}{x}=1$$

8. May 24, 2005

whozum

Doh! I retract my previous statement.