Behaviour of limits and their effect on equations

[titowakoru]

Hi,

I was wondering if some one could check my understanding of limits please.

If a limit is presetned as say x << y or x >> y am I right in thinking that x or y can be ignored as they are small enough to be insignificant? So, for example, if I had equation which was

sin(x/y)

in the situation where x << y only values of y are significant and where x >> y then only values of x are significant.

Or do I have this completely wrong??

Cheers for any advice

 

[Office_Shredder]

You have it wrong.

If you have an addition, for example 2x+3y and you are told x<<y then the x is so small that it is insignificant and you can approximate this by 3y and be OK.

But if you have a ratio like x/y, then being told x<<y doesn't let you make any simplification. If you try throwing away the x and writing it as 1/y then you're probably off by orders of magnitude which isn't a very good thing.

Sample calculations: If x<<y approximate the following:
1) $$\frac{2x + 3y}{4x - y}$$
In this case the numerator is approximately 3y and the denominator is approximately -y, so this is going to be approximately -3.

2)
$$\frac{ 2xy}{y}$$
The answer is obviously 2x. If you throw away the x because it's a lot smaller than y, you'll end up getting 2, which is not close to 2x unless x happens to be close to 1.

 

[Mark44]

I was wondering if some one could check my understanding of limits please.

If a limit is presetned as say x << y or x >> y am I right in thinking that x or y can be ignored as they are small enough to be insignificant? So, for example, if I had equation which was

sin(x/y)

in the situation where x << y only values of y are significant and where x >> y then only values of x are significant.

From what you've written, I don't think you're asking about limits, but instead are asking about how to approximate expressions in two variables, and it's given that x << y or y << x. Note that sin(x/y) is NOT an equation. An equation has an = symbol in it.

The following is an example of a limit:
$$\lim_{x \to 0}\frac{\sin (x)}{x}$$

It's possible to have limits in which a point (x, y) is approaching some fixed point (x₀, y₀), but it's very unusual for it to be given that x << y or the other way around.