mat 1234 calculus i section 1.6 part i using the limit laws
TRANSCRIPT
Quiz Friday and …
Quiz :1.6 Part I (2 problems from your HW)
Make sure you have an approved calculator.
Recall
Limit of the following form is important
1.4: Estimate limits by tables 1.6: Compute limits by algebra 1.5: Formally define limits
h
afhafh
)()(lim
0
Limit Laws
11 limit laws that help us to compute limits (printed on p.5).
Foundation of computing limits, but tedious to use.
Practical methods will be introduced.
Limit Laws
If and exist, then )(lim xfax
)(lim xgax
1. lim ( ) ( ) lim ( ) lim ( )
3. lim ( ) lim ( )x a x a x a
x a x a
f x g x f x g x
cf x c f x
Example 1
2
lim 2 5x
x
1. lim ( ) ( ) lim ( ) lim ( )
3. lim ( ) lim ( )
7. lim
8. lim
x a x a x a
x a x a
x a
x a
f x g x f x g x
cf x c f x
c c
x a
2
lim 2 5x
x
Direct Substitution Property
If is a polynomial, then
Also true if f(x) is a rational function and a is in the domain of f
)()(lim afxfax
Direct Substitution Property
If is a polynomial, then
Also true if is a rational function and is in the domain of
,
)()(lim afxfax
Direct Substitution Property
If is a polynomial, then
Also true if is a rational function and is in the domain of
)()(lim afxfax
Why?
Polynomials are “continuous” functions
lim ( ) lim ( )
m
)
l
(
i
x a x af x f x
f xx a
f a
x
y
a
( )f a
Example 2 (Rational Function, in the domain)
3 is in the domain of the rational function
2
3
6lim
5x
x
x
2
3
6lim
5x
x
x
Direct Substitution Property
Can be extended to other functions such as -th root.
Not for all functions such as absolute value, piecewise defined functions.
Limit Laws Summary
Use Direct Substitutions if possible*. That is, plug in when it is defined.
)(lim xfax
* Sums, differences, products, quotients, -th root functions of polynomials,
1.Use equal signs
2.Use parentheses for expressions with sums and differences of more than 1 term.
3. Show the substitution step.
Expectations: Standard Notations and Presentation
1
lim 1x
x
1
lim 1
1 1x
x
Expectations: Standard Notations and Presentation
4. Do not actually “cross out” terms.
1
1limx
x
1
1
x
x
Remark 1 Again
Once you substitute in the number, you do not need the limit sign anymore.
1
lim 1
1 1x
x
Remark 1 Again (What? Again!)
Once you substitute in the number, you do not need the limit sign anymore.
1
1lim
11
1 1
x x
Review: We learned…
Limit Laws Direct Substitution Property of
polynomials and rational functions Techniques
• Simplify• Combine the terms• Multiply by conjugate
Classwork
Use pencils Use “=“ signs Do not “cross out” anything. Do not skip steps
Once you substitute in the number, you do not need the limit sign anymore.