1.4 Calculating limits
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Calculating Limits - Calculus I - Lecture Slides, Slides of Calculus
In my class of Calculus-I, I take lecture note from these slides, hope these lecture slides help other student.The key point in these slides are:Calculating Limits, Limit Laws, Positive Integer, Direct Substitution Property, Rational Function, Properties of Limits, Sandwich Theorem, Squeeze Theorem, Maximum Value of Sine, Constant Multiple Law
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2012/2013
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1.4 Calculating limits
Limit Laws
Suppose that c is a constant and the limits
and exist. Then
( )
lim
f x
x โ a
( )
lim
g x
x โ a
[ ]
[ ]
[ ]
[ ]
if ( ) 0 (quotient law)
( )
( )
( )
( )
( ) ( ) ( ) ( ) (product law)
( ) ( ) (constant multiple law)
( ) ( ) ( ) ( ) (differenc elaw)
( ) ( ) ( ) ( ) (sumlaw)
lim
lim
lim
lim
lim lim
lim
lim
lim
lim lim
lim
lim lim
lim
= โ
= โ
=
โ = โ
- = +
โ
โ
โ
โ
โ โ โ
โ โ
โ โ โ
โ โ โ
g x
g x
f x
g x
f x
f x g x f x g x
c f x f x
f x g x f x g x
f x g x f x g x
x a
x a
x a
x a
x a x a x a
x a x a
x a x a x a
x a x a x a
c
Direct Substitution Property
If f is a polynomial or a rational function and a
is in the domain of f , then
Example:
lim
f x f a
x a
โ
( 2 ) 2 2 4 lim
lim lim
2
2
2
2
2
( 2 )( 2 )
2
4
- = + =
โ
โ โ
=
โ
โ +
=
โ
โ
x
x
x x
x
x x
x
x
More examples on the board.
Properties of Limits
Theorem : If f(x) โค g(x) when x is near a (except
possibly at a ) and the limits of f and g both exist
as x approaches a, then
The Squeeze Theorem : If f(x) โค g(x) โค h(x) when x
is near a (except possibly at a ) and
then
(sometimes is called the Sandwich Theorem)
lim lim
f x g x
x โ a x โ a
f x h x L
x a x a
= =
โ โ
( ) ( ) lim lim
g x L
x a
=
โ
( ) lim