9th June 2002

Basically look, say you've got a function ` y = f(x) `

,
Now y_{0} the limit of y as x tends towards x_{0},
if as we all know, "y tends to y_{0} as x tends to
x_{0}". 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.

Let ` 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^2-4) `

is
nothing but ` (x+2)(x-2) `

and we can cancel out the
` (x-2) `

in the numerator and the denominator and then
y becomes ` (x+2) `

and the limit of y as x tends to 0
is simple, its 2!

* Definition: *y_{0} is the limit of y as x tends
to x_{0}, if for any positive number epsilon, you can
always find a delta, such that whenever abs(x-x_{0}) is
smaller than delta, abs(y-y_{0}) 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 y_{0}. Then
* if I can always find a delta *, so that all numbers within
distance delta from x_{0}, have their y within epsilon from
y_{0}...and yeah, *if this is true for every such positive
epsilon*...oh forget it! "y tends to y_{0} as x tends
to x_{0}". Easy!

Lemme put it in the * mathematical style *. "If its true that

` abs(x-x`

implies that _{0}) < abs(x'-x_{0})

` abs(y-y`

_{0}) < abs(y'-y_{0})

then y_{0} is the limit".

Thats just to say that "whenever as the distance between x and
x_{0} decreases, the distance between y and y_{0} 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 y_{0}
such that y nears it as x nears x_{0} and the final distance
is smaller for such a y_{0} than any other candidate for
y_{0}." Yuck! And whats "final distance" by the way?

"whenever abs(x-x_{0}) is
smaller than delta, abs(y-y_{0}) is smaller than epsilon."
This I understand. Basically its saying that whenever x is near
x_{0}, y is near y_{0}

"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
y_{0} 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 y_{0} as x get closer to x_{0}, but
if you can find one such special y_{0}, 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 "y_{0} is the limit if y tends towards
it as x tends to x_{0} and you can get arbitrarily close to
y_{0} 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.