Chapter three Limits and Continuity

Chapter 3
Limits and Continuity

3.1 Limit of a Function
Let ( ) be defined on an open interval about except possibly at itself. We say that
the limit of ( ) as approaches is the number , and we write
lim ( ) = .
→



The Limit Laws
If , , and are real numbers,
lim ( ) = and lim ( ) = , ℎ
⟶ ⟶

1) Sum Rule: lim ( )+ ( ) = +
⟶

2) Difference Rule: lim ( )− ( ) = −
⟶

3) Product Rule: lim ( ). ( ) = . ,
⟶

4) Constant multiple Rule: lim . ( ) = . ,
⟶

5) Quotient Rule: ( )
lim = ( ≠ 0),
⟶ ( )
6) Power Rule:
lim ( ) = ( ≠ 0).
⟶


Example 3.1 Find
) lim ( +4 − 3) + −1 ) lim 4 −3
⟶ ) lim ⟶
⟶ +5
Solution

) lim ( +4 − 3) = lim + lim 4 − lim 3 = +4 −3
⟶ ⟶ ⟶ ⟶




Mathematical Analysis 25

, + − 1 lim ( + − 1) lim + lim − lim 1
⟶
) lim = = ⟶ ⟶ ⟶
⟶ +5 lim ( + 5) lim + lim 5
⟶ ⟶ ⟶

+ −1
=
+5

) lim 4 −3= lim 4 − lim 3 = 4(−2) − 3 = √16 − 3 = √11
⟶ ⟶ ⟶


Example 3.2 Evaluate
+ −2 √ + 100 − 10
) lim ) lim
⟶ − ⟶

Solution

+ −2 ( − 1)( + 2) +2 1+2
) lim = lim = lim = =3
⟶ − ⟶ ( − 1) ⟶ 1
√ + 100 − 10 √ + 100 − 10 √ + 100 + 10
) lim = lim .
⟶ ⟶ √ + 100 + 10
+ 100 − 100
= lim = lim
⟶ √ + 100 + 10 ⟶ √ + 100 + 10
1 1 1
= lim = = .
⟶ √ + 100 + 10 √100 + 10 20
Theorem 3.1. “Sandwich Theorem”
Suppose that ( ) ≤ ( ) ≤ ℎ( ) for all in some open interval containing ,
except possibly at = itself. Suppose also that
lim ( ) = lim ℎ( ) = .
⟶ ⟶

Then
lim ( ) = .
⟶




Mathematical Analysis 26