Limit of quotient of two functions

[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).

 

[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.

 

[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. :)

 

[arildno]

That's mind-reading, not parroting, uart.

 

[uart]

That's mind-reading, not parroting, uart.

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

 

[whozum]

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

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

Can a mathematician clarify?

 

[arildno]

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

 

[whozum]

Doh! I retract my previous statement.