Calculus and Vectors – How to get an A+
1.4 Limit of a Function
A Left-Hand Limit
If the values of y = f (x) can be made arbitrarily
close to L by taking x sufficiently close to a with
x < a , then:
lim− f ( x) = L
Ex 1. Use the function y = f (x) defined by the following
graph to find each limit.
x →a
Read: The limit of the function f (x) 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.
a) lim − f ( x) = DNE
x → −4
b) lim − f ( x) = 2
x →−2
c) lim − f ( x) = 2
x→−1
d) lim− f ( x) = 1
x→3
B Right-Hand Limit
If the values of y = f (x) can be made arbitrarily
close to L by taking x sufficiently close to a with
x > a , then:
lim+ f ( x) = R
x →a
Read: The limit of the function f ( x) as x
approaches a from the right is R .
Ex 2. Use the function y = f (x) defined at Ex 1. to find each
limit.
a) lim + f ( x) = 1
x→−1
b) lim+ f ( x) = ∞ ( DNE )
x→3
c) lim+ f ( x) = 3
x →1
d) lim + f ( x) = 2
x→−2
Notes:
1. R must be a number. ∞ is not a number.
2. The function may be or not defined at a .
C Limit
If the values of y = f (x) can be made arbitrarily
close to l by taking x sufficiently close to a (from
both sides), then:
lim f ( x) = l
Ex 3. Use the function y = f ( x) defined at Ex 1. to find each
limit.
a) lim f ( x) = DNE because lim − f ( x) = DNE
Read: The limit of the function f ( x) as x
approaches a is l .
c) lim f ( x) = DNE because lim+ f ( x) = ∞
x →a
x→−4
x→−4
b) lim f ( x) = DNE because lim − f ( x) = 2 ≠ lim + f ( x) = 1
x → −1
x →3
d) lim f ( x) = 1
x → −3
e) lim f ( x) = 2
x → −2
1.4 Limit of a Function
©2010 Iulia & Teodoru Gugoiu - Page 1 of 2
x →−1
x →3
x →−1
Calculus and Vectors – How to get an A+
Notes:
f) lim f ( x) = 1
x→0
1. If lim+ f ( x) = lim− f ( x) then lim f ( x) does exist
x →a
x→ a
x→a
and L = R = l .
2. If lim+ f ( x) ≠ lim− f ( x) then lim f ( x) Does Not
x →a
x→a
x→a
Exist (DNE).
3. l must be a number. ∞ is not a number.
4. The function may be or not defined at a .
Ex 4. Compute each limit.
x2
12
1
=
=
a) lim−
x→1 x + 1 1 + 1
2
12
1
x2
b) lim+
=
=
x →a
x→1 x + 1 1 + 1
2
Notes:
2
2
1
1
x
1. In order to use substitution, the function must be c) lim
=
=
x→1 x + 1 1 + 1
2
defined on both sides of the number a .
D Substitution
If the function is defined by a formula (algebraic
expression) then the limit of the function at a point
a may be determined by substitution:
lim f ( x) = f (a )
2. Substitution does not work if you get one of the
following 7 indeterminate cases:
0 ∞ ∞
∞ − ∞ 0×∞
1
∞ 0 00
0 ∞
d) lim− x − 2 = DNE
x →2
e) lim+ x − 2 = 2 − 2 = 0
x →2
f) lim x − 2 = DNE
x →2
E Piece-wise defined functions
⎧2 x − 3, x < 2
If the function changes formula at a then:
⎪
Ex 5. Consider f ( x) = ⎨0, x = 2
1. Use the appropriate formula to find first the left⎪ 2
side and the right-side limits.
⎩ x − 1, x > 2
2. Compare the left-side and the right-side limits to a) Find lim f ( x) .
conclude about the limit of the function at a .
x→2
Example:
lim− f ( x) = lim− (2 x − 3) = (2)(2) − 3 = 1
x →2
x →2
⎧ f1 ( x), x < a
f ( x) = ⎨
lim f ( x) = lim+ ( x 2 − 1) = 2 2 − 1 = 3
x →2 +
x →2
⎩ f 2 ( x), x > a
lim− f ( x) ≠ lim+ f ( x) ⇒ ∴ lim f ( x) = DNE
x →2
x →2
x →2
L = f1 (a), R = f 2 (a) (if exist)
b) Find lim f ( x) .
x→0
lim f ( x) = lim (2 x − 3) = 2(0) − 3 = −3
x →0
x →0
∴ lim f ( x) = −3
x →0
c) Draw a diagram to illustrate the situation.
5
y
4
3
2
1
x
−5
−4
−3
−2
−1
1
−1
−2
−3
−4
Reading: Nelson Textbook, Pages 34-37
Homework: Nelson Textbook: Page 37 #4d, 5, 6, 7, 10cef, 11c, 15
1.4 Limit of a Function
©2010 Iulia & Teodoru Gugoiu - Page 2 of 2
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