MAT 1234
Calculus I
Section 2.3
Basic Limit Laws
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HW
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Homework 2.3
Do your HW ASAP.
Tutoring is available!!!
Seriously, Do not wait.
Quiz Friday and …
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Quiz: Everything up to 2.5
(A Maple question?)
Make sure you have an approved
calculator.
Recall
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
2.1: Estimate limits by tables
2.5: Compute limits by algebra
2.2: Formally define limits
Preview
 Limit
Laws
 Direct Substitution Property
 Practical summary of all the limit laws
Limit Laws
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11 limit laws that help us to compute
limits (printed on p.5, numbering slightly
different from the textbook).
Foundation of computing limits, but
tedious to use.
Practical methods will be introduced.
Limit Laws
7. lim c  c
y
xa
yc
c
a
x
Limit Laws
8. lim x  a
y
x a
yx
a
x
Limit Laws
If
lim f ( x )
x a
g ( x) exist, then
and lim
x a
1. lim  f ( x)  g ( x)   lim f ( x)  lim g ( x)
xa
xa
3. lim  cf ( x)   c lim f ( x)
xa
x a
x a
Example 1
lim  2 x  5
x 2
1. lim  f ( x)  g ( x)   lim f ( x)  lim g ( x)
xa
3. lim  cf ( x)  c lim f ( x)
xa
7. lim c  c
xa
8. lim x  a
xa
lim  2 x  5 
x2
xa
xa
x a
Direct Substitution Property
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If 𝑓(𝑥) is a polynomial, then
lim f ( x)  f (a)
xa
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Also true if f(x) is a rational function and
a is in the domain of f
Direct Substitution Property
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If 𝑓(𝑥) is a polynomial, then
lim f ( x)  f (a)
xa
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Also true if 𝑓(𝑥) is a rational function and
𝑎 is in the domain of 𝑓
𝑓 𝑥 =
𝑥−5
,
𝑥+5
𝑎=0
Direct Substitution Property
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If 𝑓(𝑥) is a polynomial, then
lim f ( x)  f (a)
xa
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Also true if 𝑓(𝑥) is a rational function and
𝑎 is in the domain of 𝑓
Why?
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Polynomials are “continuous” functions
y
lim f ( x)  f (a)
xa
x
a
Why?
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Polynomials are “continuous” functions
y
lim f ( x)  lim f ( x)  f (a)
x a 
x a
f (a)
a
x
Why?
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Polynomials are “continuous” functions
y
lim f ( x)  lim f ( x)  f (a )
xa
xa
lim f  x 
xa
f (a)
a
x
Example 1 (Polynomial)
lim  2 x  5
x 2
lim  2 x  5 
x2
Remark 1
Once you substitute in the number, you do
not need the limit sign anymore.
Example 2
(Rational Function, 𝑎 in the domain)
x2  6
lim
x 3  x  5
Example 2
(Rational Function, 𝑎 in the domain)
x2  6
lim
x 3  x  5
x2  6
lim

x 3  x  5
3 is in the domain of
the rational function
Direct Substitution Property
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Can be extended to other functions such
as 𝑛-th root.
Not for all functions such as absolute
value, piecewise defined functions.
Limit Laws Summary
lim f ( x )
x a
Use Direct Substitutions if possible*.
That is, plug in 𝑥 = 𝑎 when it is defined.
* Sums, differences, products, quotients, 𝑛-th root
functions of polynomials,
Example 3
lim 3 x3  x2  8
x 1
lim
x 1
3
x  x 8 
3
2
Q&A
Q: What to do if the answer is undefined
when plugging in 𝑥 = 𝑎?
x 1
lim
x 1 x  1
2
A: We will look at algebraic techniques in
section 2.5
Review: We learned…
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Limit Laws
Direct Substitution Property of
polynomials and rational functions
Classwork
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Further Explorations of Continuous
Functions
Definition
A function 𝑓 is continuous at a point 𝑎 if
lim f ( x)  f (a)
xa
That means…
𝑎 lim 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠
𝑥→𝑎
𝑏 𝑓 𝑎 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑐 lim 𝑓 𝑥 = 𝑓(𝑎)
𝑥→𝑎
Example
For what value of 𝑐 is the function 𝑓
continuous at 𝑥 = 2?
 cx  1
f ( x)   2
cx  1
Demo
if x  2
if x  2