First of all, it is not true that 
is 
. In fact, this is never true (note that 
is undefined). For example:
.
.And so on.
What is true is that 
but this doesn't say that there is a specific value of 
such that 
; rather, it says intuitively that by picking 
really really close to 
we can make 
really really close to 
.
The precise definition of the limit is a bit more complicated: when we say 
what we mean is that for any 
(the "degree of accuracy") we can guarantee that 
is within 
of 
just by making sure that 
is close enough to 
; fully formally, that for all 
there is some 
such that 
(There is a slight asymmetry here, namely the "
" on the 
-side but not on the 
-side, but that's best ignored at first.)
Re: your other question, this is a specific property about 
. For example, 
is undefined (from the left it approaches 
and from the right it approaches 
).
First of all, it is not true that is . In fact, this is never true (note that is undefined). For example:
- .
- .
And so on.
What is true is that but this doesn't say that there is a specific value of such that ; rather, it says intuitively that by picking really really close to we can make really really close to .
The precise definition of the limit is a bit more complicated: when we say what we mean is that for any (the "degree of accuracy") we can guarantee that is within of just by making sure that is close enough to ; fully formally, that for all there is some such that (There is a slight asymmetry here, namely the "" on the -side but not on the -side, but that's best ignored at first.)
Re: your other question, this is a specific property about . For example, is undefined (from the left it approaches and from the right it approaches ).
One way to think about this is to recall the relationship between the trig functions and circles (in particular and wlog, the unit circle).
First, we're measuring angles by arc length subtended (i.e., in radians). The theorem comes from the fact that when the angle is very small (call it 
), then 
(draw a diagram!). More correctly, though, what holds is that 
for 
, so that we have, upon division by 
and taking reciprocals, that 
Now letting 
, we obtain the result (a similar argument can be made from the other direction).
No, it doesn't hold for the other trig functions similarly. In particular, 
I've been trying to prove that the derivative of sin is cos, and have managed to prove using the definition of limit that the derivative of sin(x) is cos(x) * lim[h->0] sin(h)/h; I know the limit is 1, but I can't prove it.
I can think of ways to prove the limit using taylor series and l'hopital's rule (also I can't use l'hopital's b/c I havent proved this), but not without accepting that the derivative of sin(x) = cos(x) in the first place.
To be clear- I DONT WANT TO KNOW how to actually prove the thing, I just want to know if a method exists which I could use having only taken AP Calc
I'm having some trouble with the intuition with limits.
lim h->0 (cos(h)-1)/h = 0. But how do you reason to arrive at that? I could just plug in numbers and see what it approaches but it feels like a weak strategy. I know that cos(0) = 1. This means that the numerator will approach 0. So will the demoninator. But is that enough to say that the limit is 0? If it is, then the next limit is a little troubling.
lim h->0 sin(h)/h = 1. If we use the same reasoning as above and given that sin(0) = 0, then we have something approaching 0 in the numerator and something approaching 0 in the denominator. Quite the same situation as above with the cos expression. But the real answers to the limits are not equal. In one case it's 0 and in the other it's 1.
I'm feeling a little lost in how to reason about this. Any help is appreciated.