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Transcript of 14_Limit_of_a_Function.pdf
8/10/2019 14_Limit_of_a_Function.pdf
http://slidepdf.com/reader/full/14limitofafunctionpdf 1/2
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
8/10/2019 14_Limit_of_a_Function.pdf
http://slidepdf.com/reader/full/14limitofafunctionpdf 2/2
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