Topic today: Limit computation (a practical lesson)
More (non-)existent criteria for limit.
Continuity
Limit computation (from basic building blocks)
Review: Theorem 2.2 relating one-sided and two-sided limits. Using this thm to confirm the (non-)existence of a limit.
Two more general criteria guaranteeing the existence of limit:
(a) monotonic and bounded;
(b) the squeezing principle.
Let us explain in detail.
(a) increasing with an upper bound; decreasing with a lower bound.
Increasing with an upper bound: 
as 
increasing, 
also increases.
What is a upper bound? It is a (finite) constant (say 
), such that the thing we are considering (here is the value of ) is always less than (or equal to ) . (in the example , we can take 
).
Symbolically, 
%20%3D%200#card=math&code=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%281-e%5E%7B-x%7D%29%20%3D%200).
Remark: the upper bound is of course not unique. (any number bigger than is still an upper bound).
The criterion of “decreasing with a lower bound” is totally symmetric.
decreasing with a lower bound 
. 
.
(
) is also decreasing with a lower bound . 
%20%3D%200#card=math&code=%5Clim_%7Bx%5Cto%20%5Cinfty%20%7D%281%2Fx%29%20%3D%200).
There are examples 
%20%3D%20L%20%3C%20%5Cinfty#card=math&code=%5Clim_%7Bx%20%5Cto%20x_0%7D%20f%28x%29%20%3D%20L%20%3C%20%5Cinfty) where 
.
(b) the “squeezing” principle. (The squeeze theorem 2.7)
The typical application of the squeeze theorem is: 
.
(See Equation (2.18) on p. 181 of the OpenStax Calculus book.)
Application of the criterion (a): Explanation (not strict proof !) of 
when 
.
(1) Trivial if 
(see Theorem 2.4, Equation (2.15)); So suppose 
.
(2) if 
, then 
decreases as 
, with lower bound .
(3) if 
, consider the two sequences 
and 
.
%5En#card=math&code=q%5E%7B2n%7D%20%3D%20%28q%5E2%29%5En) just as discussed in (2); 
, increasing with upper bound .
We can boldly guess that if then 
as .
- Notion of continuous functions
In particular, three types of discontinuities:
(a) removable;
(b) jump;
(c) infinite.
(Enhanced) Theorem: Let 
be an elementary function (that is, 
of basic elementary functions, namely, polynomials, exponential, logarithmic, trigonometric, and inverse of trigonometric functions) defined over 
%20%5Ccup%20%5Ccdots%20%5Ccup%20(an%2C%20b_n)#card=math&code=I%20%3D%28a_1%2C%20b_1%29%20%5Ccup%20%5Ccdots%20%5Ccup%20%28a_n%2C%20b_n%29), where .
Conclusion: Then is continous over any open interval 
#card=math&code=%28a_i%2C%20b_i%29).
(See Theorem 2.8 in the OpenStax Calculus book)
What about 
%3D%5Csin(x)%2Fx#card=math&code=g%28x%29%3D%5Csin%28x%29%2Fx)? What is the value of 
#card=math&code=g%280%29)? (Answer: 
is not inside the domain of #card=math&code=g%28x%29).)
- Application: Limit computation!
Using Theorem 2.4, 2.5 (limit laws) + the above enhanced version of Theorem 2.8, 2.10 and Theorem 2.9, we can compute a lot of limits.
Examples:
(1). 
#card=math&code=%5Ccos%20%28x%20%20-%20%5Cpi%2F2%29): we can regard it as the composition 
%2C%20u%20%3D%20u(x)%3Dx-%20%5Cpi%2F2#card=math&code=y%3D%5Ccos%20%28u%29%2C%20u%20%3D%20u%28x%29%3Dx-%20%5Cpi%2F2). At 
, both 
#card=math&code=y%3D%5Ccos%28u%29) and 
%3Dx-%5Cpi%2F2#card=math&code=u%28x%29%3Dx-%5Cpi%2F2) are continuous. So just insert 
into the expression: 
%3D0#card=math&code=u%28%5Cpi%2F2%29%3D0). and 
%20%3D%20%5Ccos(0)%20%3D%201#card=math&code=y%280%29%20%3D%20%5Ccos%280%29%20%3D%201) . So 
%3D1#card=math&code=%5Clim_%7Bx%5Cto%20%5Cpi%2F2%7D%20%5Ccos%28x-%5Cpi%2F2%29%3D1).
(2). 
%2F(x%5E2%2Bx-6)#card=math&code=%5Clim_%7Bx%5Cto%202%7D%20%28x%5E2-4%29%2F%28x%5E2%2Bx-6%29). When 
, both 
and 
are zero.
We observe that 
%3D(x-2)(x%2B2)#card=math&code=%28x%5E2-4%29%3D%28x-2%29%28x%2B2%29) , 
%3D(x-2)(x%2B3)#card=math&code=%28x%5E2%2Bx-6%29%3D%28x-2%29%28x%2B3%29).
(x%2B2)%7D%7B(x-2)(x%2B3)%7D%20%3D%20%5Clim%7Bx%5Cto%202%7D%20%5Cfrac%7Bx%2B2%7D%7Bx%2B3%7D%20%3D%20%5Cfrac%7B2%2B2%7D%7B2%2B3%7D%3D4%2F5#card=math&code=%5Clim%7Bx%20%5Cto%202%7D%20%5Cfrac%7B%28x-2%29%28x%2B2%29%7D%7B%28x-2%29%28x%2B3%29%7D%20%3D%20%5Clim_%7Bx%5Cto%202%7D%20%5Cfrac%7Bx%2B2%7D%7Bx%2B3%7D%20%3D%20%5Cfrac%7B2%2B2%7D%7B2%2B3%7D%3D4%2F5).
