L ecture2| 1 Chapter 1
Limits and Continuity
Outline 5. Limit Laws 6. Continuity
L ecture2| 2 5. Limit Laws
We had formulated the idea that using tables or graphs.
The value is called the limit and is denoted
However, we need a quick and precise way to calculate the limits.
L ecture2| 3 Limit Laws
Suppose is a constant, are functions such that
Then
1. (Sum)
2. (Subtraction)
3. (Constant multiplication)
L ecture2| 4 4. (Product)
5. (Quotient)
provided
Remark The limit laws are easily seen to be true. For example, if is close to and is closed to , then is close to . So it is intuitively clear that the sum law is true.
L ecture2| 5 6. (Power)
where
7. (Root)
where If (even), we assume that .
L ecture2| 6
8. (Special)
and
where If (even) we assume .
L ecture2| 7 EX Evaluate the limit
Remark Every polynomial
has the direct substitution property at every , i.e.
L ecture2| 8 EX Evaluate the limit
Remark Every rational function
has the direct substitution property at every such that , i.e.
provided .
L ecture2| 9 EX Evaluate the limits
L ecture2| 10 EX (The limit laws are true for one-
sided limits) Evaluate the limits
Remark In (3), the function
stays positive as , so the root law is true.
L ecture2| 11 EX (Do some preliminary algebra)
Evaluate the limit
L ecture2| 12 EX Evaluate the limits
Remark
L ecture2| 13 EX (Absolute value functions) Evaluate the limit
by simplifying the function
as .
L ecture2| 14
Remark
For finding a limits with , we can assume and close to .
For finding a limit with , we can assume and close to .
L ecture2| 15 EX Show that the limit
does not exist.
L ecture2| 16 EX (Rationalization) Evaluate the limit
Remark
L ecture2| 17 EX Find
L ecture2| 18
Theorem (Squeeze)
If when is close to and
then
Proof
L ecture2| 19 EX Let be a function such that
for close to .
Evaluate the limit
L ecture2| 20 EX Show that
by verifying that
Remark
L ecture2| 21 6. Continuity
Def A function is continuous at if
I.e. the DSP is true for at .
If is not continuous at , we say that is discontinuous at .
L ecture2| 22 To be continuous at , all the
following conditions must be true:
Failing to satisfy at least one of the
above conditions implies that the function is discontinuous at .
L ecture2| 23 EX Determine whether the function
is continuous at ? Find the set of all points where is continuous.
L ecture2| 24 EX For the following graph of a function, is the function continuous at ?
L ecture2| 25 EX Explain why the function is
discontinuous at the given number ?
L ecture2| 26
Def A function is continuous from the right at if
and is continuous from the left at if
L ecture2| 27
Def is continuous on if is cont.
at every .
is continuous on if is cont. on and it is cont. from the right at .
is continuous on if is cont. on and it is cont. from the left at .
is continuous on if is cont. on , is cont. from the right at , and is cont. from the from the left at .
L ecture2| 28 EX Show that
is continuous on .
L ecture2| 29 EX Show that
is continuous on .
L ecture2| 30
Rule 1 Assume and are continuous at . Let be a constant.
The following functions are cont. at :
and
Remark The rule is true by the limit laws.
For example, since are continuous at , it follows that is close to and is close to . So is close to .
L ecture2| 31
Rule 2 Functions which are continuous at every number in their domains:
1. Polynomials
2. Rational functions
3. Power functions
4. Root functions If , the function is continuous on .
5. Trig functions
L ecture2| 32 EX Find the domain for each of the
following functions. Explain why they are continuous on their domains?
(1)
(2)
(3)
L ecture2| 33 EX (Use the continuity to find limits)
Evaluate
L ecture2| 34
Def Let and be functions. Define the function by
is called the composite function of and . It is denoted by
L ecture2| 35 EX with
with
L ecture2| 36
Limits of composite functions
If has the limit as and is a continuous function, then
L ecture2| 37 EX Evaluate the limit
where
L ecture2| 38
Continuity of composite functions If is continuous at and is continuous at , then the composite function is continuous at .
Remark This fact follows directly from the definition. In fact, since is cont. at and is cont. at , we get is close to as and is close to
as . So is close to . Thus is cont. at .
L ecture2| 39 EX Where are the following functions
continuous?
(a) (b)
