Topic today: The limit
What is a limit?
When does a limit exists?
If not existed, anything we can do to save the game?
Applications of limit.
Why is 0.999999999….. = 1?
And the definition of Euler’s e,
- Given a function
#card=math&code=y%3Df%28x%29), when 
approaches to a fixed point (a constant) 
from both the left and the right hand-sides, if 
#card=math&code=f%28x%29) approaches to a value 
, then we say the limit of #card=math&code=f%28x%29) is .
%20%3D%201#card=math&code=g%28x%29%20%3D%201), if 
; %3D-1#card=math&code=g%28x%29%3D-1) if 
. But #card=math&code=g%28x%29) has no definition at 
.
%20%3D%20(x-2)%5E%7B-2%7D#card=math&code=h%28x%29%20%3D%20%28x-2%29%5E%7B-2%7D), this is similar to 
%20%3D%20-1%2Fx%5E2#card=math&code=y%28x%29%20%3D%20-1%2Fx%5E2).
The function #card=math&code=f%28x%29) at the beginning of section 2.2, and the function #card=math&code=g%28x%29) in example 2.6, have a common thing: at the point where we want to find out the limit, the function has no definition, in other words, the graph of the function is broken at the point.
Notation: 
or 
, means (the variable) approaching to 
from right and left, respectively.
- The notion of one-sided limit provides a more accurate answer to the behaviour of a function around a point which is not in the domain of the function.
Modification of Example 2.8 (one-sided limits often appear in piecewise functions)
%3Dx%2B1#card=math&code=F%28x%29%3Dx%2B1) if 
, and %3Dx%5E2-4#card=math&code=F%28x%29%3Dx%5E2-4) if 
.
%3D3#card=math&code=F%282%29%3D3), while 
%3D3#card=math&code=%5Clim_%7Bx%20%5Cto%202%5E-%7D%20F%28x%29%3D3).
Compare to the #card=math&code=f%28x%29): 
%3D0#card=math&code=f%282%29%3D0), and it is just 
%3D0#card=math&code=%5Clim_%7Bx%20%5Cto%202%5E%2B%7Df%28x%29%3D0).
We can modify even worse:
%20%3D%20x%2B1#card=math&code=f_1%28x%29%20%3D%20x%2B1) if , %3Dx%5E2-4#card=math&code=f_1%28x%29%3Dx%5E2-4) if 
, and 
%20%3D%20100#card=math&code=f_1%282%29%20%3D%20100).
%20%3D%203#card=math&code=%5Clim%7Bx%20%5Cto%202%5E-%7D%20f_1%28x%29%20%3D%203), %20%3D%200#card=math&code=%5Clim%7Bx%5Cto%202%5E%2B%7D%20f_1%28x%29%20%3D%200). Neither of these one-sided limits equals to #card=math&code=f_1%282%29).
Remark: Theorem 2.2 which relates one-sided and two-sided limits, is usually used to prove the non-existence of a limit for certain functions, for example, the g(x) at the beginning of section 2.2, or the limit 
, as well as the Example 2.7 
#card=math&code=%5Clim_%7Bx%5Cto%200%7D%20%5Csin%281%2Fx%29).
Now review the three functions at the beginning of Section 2.2.
- Application:
We first mention without proof of a famous limit: 
if 
.
%20%3D%209(%5Cfrac%7B1%7D%7B10%7D%2B%5Cfrac%7B1%7D%7B100%7D%20%2B%20%5Cfrac%7B1%7D%7B1000%7D%2B%20…)%20%3D%209%20%5Ctimes%20%5Cfrac%7B1%7D%7B9%7D%3D1#card=math&code=0.99999999%E2%80%A6%20%3D%209%280.1%2B0.01%2B0.001%2B%E2%80%A6%29%20%3D%209%28%5Cfrac%7B1%7D%7B10%7D%2B%5Cfrac%7B1%7D%7B100%7D%20%2B%20%5Cfrac%7B1%7D%7B1000%7D%2B%20…%29%20%3D%209%20%5Ctimes%20%5Cfrac%7B1%7D%7B9%7D%3D1).
, 
%2F(1-q)#card=math&code=Sn%20%3D%20a_1%2Ba_2%20%2B%20%5Cldots%20%2B%20a_n%20%3D%20a_1%281-q%5E%7Bn%7D%29%2F%281-q%29) where 
. %2F(1-q)%20%3D%20%5Cfrac%7Ba_1%7D%7B1-q%7D%20%3D%201%2F10%20%2F%209%2F10%20%3D%201%2F9#card=math&code=%5Clim%7Bn%20%5Cto%20%5Cinfty%7D%20Sn%20%3D%20%5Clim%7Bn%20%5Cto%20%5Cinfty%7D%20a_1%20%281-q%5En%29%2F%281-q%29%20%3D%20%5Cfrac%7Ba_1%7D%7B1-q%7D%20%3D%201%2F10%20%2F%209%2F10%20%3D%201%2F9).
What is the exact value of 
? (
approximately)
More general question: why do me express one real number in different form?
We will discuss this question at Thursday’s lectures.
