9th June 2002
Basically look, say you've got a function
y = f(x) ,
Now y0 the limit of y as x tends towards x0,
if as we all know, "y tends to y0 as x tends to
x0". Very easy to understand intuitively!
Okay, consider y = x + 2 for instance, the limit of y as x tends towards 0, is obviously 2.
Here's an example of "tricky problems" that we could easily tackle.
y = (x^2 - 4) / (x-2).
Now find the limit of y as x tends to 2.
If you substitute 2 for x, the denominator becomes 0 and all hell
breaks lose! But we're too smart for that.
(x+2)(x-2) and we can cancel out the
(x-2) in the numerator and the denominator and then
(x+2) and the limit of y as x tends to 0
is simple, its 2!
Definition: y0 is the limit of y as x tends to x0, if for any positive number epsilon, you can always find a delta, such that whenever abs(x-x0) is smaller than delta, abs(y-y0) is smaller than epsilon.
Huh? Well yeah, whatever. It probably means the same thing, that I've understood.
Can't get that definition off my mind. Okay, let me just think it out. First I fix an epsilon - a distance for y from y0. Then if I can always find a delta , so that all numbers within distance delta from x0, have their y within epsilon from y0...and yeah, if this is true for every such positive epsilon...oh forget it! "y tends to y0 as x tends to x0". Easy!
Lemme put it in the mathematical style . "If its true that
abs(x-x0) < abs(x'-x0)
abs(y-y0) < abs(y'-y0)
then y0 is the limit".
Thats just to say that "whenever as the distance between x and x0 decreases, the distance between y and y0 decreases."
Okay, now off to sleep.
y = x^2. I know that the limit of y as x tends to 0 is 0. Does y near 0 as x nears 0? Yes, cool.
Hey, but by the same token, can't you say -1 is the limit? Doesn't the distance between y and -1 decrease as the distance between x and 0 decreases. Oh no! So anything smaller than 0 is a limit by my definition.
Let me try again. So whats special about 0 then? Well finally I can see that it tends towards 0! Not -1 obviously! "The limit is a number y0 such that y nears it as x nears x0 and the final distance is smaller for such a y0 than any other candidate for y0." Yuck! And whats "final distance" by the way?
"whenever abs(x-x0) is smaller than delta, abs(y-y0) is smaller than epsilon." This I understand. Basically its saying that whenever x is near x0, y is near y0
"for any positive epsilon, you can always find a delta". Yeah this is the part that achieves the magic. For any positive epsilon. Epsilon can be very small, very big. Very small. The distance between y and y0 can be very very little.
Got it! Its all light! I am a genius! What it says is that, not only does y get closer to y0 as x get closer to x0, but if you can find one such special y0, to which you can get as close as you want (for any positive epsilon, however small), and that is the limit!
Yeah! That is why 0 is the limit of
y = x^2 and not -1.
Yes, y nears -1 as x nears 0, but you can't get as close as
you want to -1 by taking x towards 0, you can get that close
only to 0.
So from now on I'll say "y0 is the limit if y tends towards it as x tends to x0 and you can get arbitrarily close to y0 this way". Simple!
Phew! Now I can sleep in peace! Zzzzzzzzz.....
I should not neglect any part of whatever I learn, and not presume what it means.
There are many subtleties in many concepts that you definitely have a feel of, but have difficulty in pinpointing.
To check whether I have understood anything properly, I should try and see whether I can explain it to somebody else (or to myself pretending to be somebody else).
There's a lot to learn from all this, but well this is all I'll write about - I know you get the idea.