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Calculus and Vectors – How to get an A+

1.4 Limit of a Function©2010 Iulia & Teodoru Gugoiu - Page 1 of 2

1.4 Limit of a Function

A Left-Hand LimitIf the values of )( x f y = can be made arbitrarilyclose to L by taking x sufficiently close to a with

a x < , then: L x f

a x=

−→)(lim

Read: The limit of the function )( x f as x approaches a from the left is L .

Notes:

1. The function may be or not defined at a .2. DNE stands for Does Not Exist .3. L must be a number.4. ∞ is not a number.

Ex 1. Use the function )( x f y = defined by the followinggraph to find each limit.

a) DNE x f x

=−−→

)(lim4

b) 2)(lim2 =−−→ x f x c) 2)(lim

1=

−−→ x f

x

d) 1)(lim3

=−→

x f x

B Right-Hand LimitIf the values of )( x f y = can be made arbitrarilyclose to L by taking x sufficiently close to a with

a x > , then: R x f

a x=

+→)(lim

Read: The limit of the function )( x f as x

approaches a from the right is R .

Notes:1. R must be a number. ∞ is not a number.2. The function may be or not defined at a .

Ex 2. Use the function )( x f y = defined at Ex 1. to find eachlimit.a) 1)(lim

1=

+−→ x f

x

b) )()(lim3

DNE x f x

∞=+→

c) 3)(lim1

=+→

x f x

d) 2)(lim2 =+−→ x f x

C LimitIf the values of )( x f y = can be made arbitrarilyclose to l by taking x sufficiently close to a (fromboth sides), then:

l x f a x

=→

)(lim

Read: The limit of the function )( x f as x approaches a is l .

Ex 3. Use the function )( x f y = defined at Ex 1. to find each

limit.a) DNE x f x

=−→

)(lim4

because DNE x f x

=−−→

)(lim4

b) DNE x f x

=−→

)(lim1

because 1)(lim2)(lim11

=≠=+− −→−→

x f x f x x

c) DNE x f x

=→

)(lim3

because ∞=+→

)(lim3

x f x

d) 1)(lim3

=−→

x f x

e) 2)(lim2

=−→

x f x

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Calculus and Vectors – How to get an A+

1.4 Limit of a Function©2010 Iulia & Teodoru Gugoiu - Page 2 of 2

Notes:1. If )(lim)(lim x f x f

a xa x −+ →→= then )(lim x f

a x→does exist

and l R L == .2. If )(lim)(lim x f x f

a xa x −+ →→≠ then )(lim x f

a x→Does Not

Exist (DNE).3. l must be a number. ∞ is not a number.

4. The function may be or not defined at a .

f) 1)(lim0

=→

x f x

D Substitution If the function is defined by a formula (algebraicexpression) then the limit of the function at a pointa may be determined by substitution :

)()(lim a f x f a x

=→

Notes:1. In order to use substitution, the function must bedefined on both sides of the number a .2. Substitution does not work if you get one of thefollowing 7 indeterminate cases :

00010

00 ∞∞

∞∞×∞−∞

∞

Ex 4. Compute each limit.

a)21

111

1lim

22

1=

+=

+−→ x x

x

b)21

111

1lim

22

1=

+=

++→ x x

x

c)21

111

1lim

22

1=

+=

+→ x x

x

d) DNE x x

=−−→

2lim2

e) 0222lim2

=−=−+

→ x

x

f) DNE x x

=−→

2lim2

E Piece-wise defined functions If the function changes formula at a then:1. Use the appropriate formula to find first the left-side and the right-side limits.2. Compare the left-side and the right-side limits toconclude about the limit of the function at a .Example:

⎩⎨⎧

><

=a x x f

a x x f x f

),(

),()(

2

1

)(),( 21 a f Ra f L == (if exist)

Ex 5. Consider⎪⎩

⎪⎨

⎧

>−

=<−

=

2,1

2,0

2,32

)(2 x x

x

x x

x f

a) Find )(lim2

x f x→

.

DNE x f x f x f

x x f

x x f

x x x

x x

x x

=∴⇒≠

=−=−=

=−=−=

→→→

→→

→→

+−

++

−−

)(lim)(lim)(lim

312)1(lim)(lim

13)2)(2()32(lim)(lim

222

22

22

22

b) Find )(lim0

x f x→

.

3)(lim

33)0(2)32(lim)(lim

0

00

−=∴

−=−=−=

→

→→

x f

x x f

x

x x

c) Draw a diagram to illustrate the situation.

−5 −4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

5

x

y

Reading : Nelson Textbook, Pages 34-37Homework : Nelson Textbook: Page 37 #4d, 5, 6, 7, 10cef, 11c, 15