Limit of ratio vs. Ratio of limit

[MDJH]

Ok, so supposedly in calculus it's a rule that the limit of a ratio of two functions should be logically equivalent to the ratio of the limits of those two functions. Supposedly even for discrete sequences the same applies to limits at infinity.

 

So what if two functions (or sequences) tend to infinity on their own? Let's say we have f(x) and g(x), both of which tend to infinity as x tends to infinity... however, the limit at infinity of f(x)/g(x) is a finite number, let's say c.

 

If c were greater than 1, would it be logically equivalent to say that lim f(x) > lim g(x) even though both tend to infinity?

 

Similarily, if c were less than 1, would it be logically equivalent to say that lim f(x) < lim g(x) even though both tend to infinity?

Edited July 27, 2010 by MDJH

[timo]

I think you should look up the rule to see if it exists at all. Then check what conditions are set on the functions, e.g. if they have to have a real(-valued), non-zero limit. The way you stated it, it is obviously wrong.

Edited July 28, 2010 by timo

[Danijel Gorupec]

MDJH,

 

I don't think you can use standard inequality signs ('<' and '>') to compare such 'infinte' kind of numbers. These signs are not to be used this way.

 

However, as math is just a construction, you can define your own operators 'less than' and 'greater than' and you can choose to use '<' and '>' symbols for them. Then you can define that 'lim f(x) > lim g(x)' if (and only if) 'lim f(x)/g(x)' gives real number greater than 1... There are no internal problems with this definition, so math alowes it... However, it would be nice if you can also find a good use for this math.

[Shadow]

Don't you apply L'Hopital if you get [math]\frac{\infty}{\infty}[/math]?

 

EDIT: Never mind, I missed the point.

Edited July 27, 2010 by Shadow