(Original post by **99wattr89**)

At least now I understand why I don't understand! xD

Thank you so much for sticking with me to explain.

So, it seems that what I don't understand is how we can change h's value after simplification. It has to be =/= 0 for the simplification, but then becomes 0 after?

Also, in the derivation in your first post, h seems to have to be 0 for the simplification itself to work.

I'm trying to find a way to explain limits without them being confusing, but if you've just done C3 (or even C4) then it's hard to put it in terms you'll understand off the bat... this stuff isn't normally covered until first year of uni.

In cases like these, letting

tend to zero is like looking at the limit of a sequence or a series. Say for example we have a geometric series

with initial term 1 and common ratio

.

The formula

reveals that

This means that if we add an infinite number of these terms to each other, we end up with 2. However, it's obviously physically impossible to add together an infinite number of terms; and this is why we say that 2 is the 'limit' of the series. When we deal with things we can't physically do (e.g. divide by zero, add infinite things together, and so on), we have to appeal to limits.

When we take a limit when adding an infinite number of things together, we look at what happens when we add a finite number of things together, as the finite number grows. So with the geometric progression above, we look at the nth partial sum

. Now we can derive that

...and to find the infinite limit, we see what happens as

grows; but notice that all this time

is still finite. We end up with the above boxed result when we see its behaviour as it tends to infinity. The key word here is "tends to".

We're doing the same thing with

here. We can't let it actually

__be__ zero, but we can look at what happens when it's

__close to__ zero. In particular, when it's

__arbitrarily close to__ zero. I won't go too far into what we mean by "arbitrarily close to", because that's a whole different ball game, but essentially this is the reason why we can treat h as if it's not zero before we take the limit.