(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
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.