2.1 FINITE LIMITS One Sided Limits, Double Sided Limits and Essential Discontinuities Mathgotserved.com.
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2.1 FINITE LIMITS
One Sided Limits, Double Sided
Limits and Essential Discontinuities
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Geometric View of Limits
• Look at a polygon inscribed in a circle
As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.
If we refer to the polygon as an n-gon, where n is the number of sides we can make
some mathematical statements:
• As n gets larger, the n-gon gets closer to being a circle
• As n approaches infinity, the n-gon approaches the circle
• The limit of the n-gon, as n goes to infinity is the circle
lim( )n
n go circlen
The symbolic statement is: Note: The n-gon approaches a circle in appearance even though it is not a circle. It might as well be a circle.
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Definition (verbal)
The limit of a function is the value the function approaches as the independent variable approaches a value, negative or positive infinity. Or think of it as the y value the function approaches as you approach a specific x value.
The Limit of a Function
lim
x a
f x L
Analytical Representation
“Read as the limit of f(x) as x approaches a
is L”Can also be expressed as f(x)→L
as x→a
Properties of Limits
lim lim lim
x c x c x c
f x g x f x g x1. Sum Rule:
lim lim lim
x c x c x c
f x g x f x g x2. Difference Rule:
lim lim limx c x c x c
f x g x f x g x
3.Product Rule:
limlim
lim
x c
x cx c
f xf x
g x g x4. Quotient
Rule:
lim
x c
k k5. Constant Rule:
lim lim
rr ss
x c x cf x f x6. Power Rule
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WHY LEARN LIMITS?
Limits are used in derivatives when mathematicians are calculating the velocity of an object in flight.
Real World Application : Calculating Speed(Mathematicians: 42-145k/yr)
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One-Sided Limits (Verbal)Numbers x near c fall into two natural categories: those that lie to the left of c and those that lie to the right of c. We write[The left-hand limit of f(x) as x tends to c is L.]
to indicate that as x approaches c from the left, f(x) approaches L.
[The right-hand limit of f(x) as x tends to c is L.]
to indicate that as x approaches c from the right, f(x) approaches L
limx c
f x L
limx c
f x L
We write
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How close can Raul Get to the garage?
limraul wall
distance from wall = 0
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Definition of a Limit
Left hand Limit:
We say the limit as x approaches a from the left is
limx a
Lf x
L
x
y
L --
| a
We say the limit as x approaches a from the left is
limx a
Lf x
Right hand Limit:
Double sided Limits
L
limx a
f x L
We say the limit as x approaches a is L
x a x a
Procedures for Finding Finite Limits Algebraically
1. Establish the fact that the limit is finite. (i.e The value that x and the f(x) approaches must not be )
2. Substitute the value x is approaching and evaluate. If 0 is in the denominator go to 3.
3. Factor, or rationalize the numerator or denominator and cancel out any removable discontinuities and substitute again. If 0 at the bottom use a non algebraic approach since the limit might not exist.
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Finding Finite Limits Algebraically
3 2
11. lim 3 5
xx x
Determine the limit of the following
3 23 1 5 1 2
21
12. lim
1x
x
x
1
1 1
x
x x1
x
1x 1x
1
1
x
1
1
1
13. .lim
1
1.lim
1
1.lim
1
x
x
x
xa
x
xb
x
xc
x
1
1
x
x1
1
1
x
x1
1
1
x
x 1 Does not exist
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0
sin 3limx
x
x
Problem 4:Determine the limit
Solution:0
sin 3limx
x
x
0
sinlim 1x
x
x
sin3
3
3x
x
0
sin 3lim3
3
x
x
x
0
sin 33lim
3
x
x
x
3 1 3
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0
sin 2lim
sin 3x
x
x
Problem 5:Determine the limit
Solution:
0
sinlim 1x
x
x
0
sin 2lim
sin 3x
x
x 0
sin 2lim
sin
2
3
2
33
x
xxxxx
x0
sin 22
2lim
sin 33
3
x
xx
xx
xx
2
x
3 x 0
sin 22
limsin 3
3
x
xx
xx
0
0
sin 2lim
2 2sin 33
lim3
x
x
xx
xx
2 1
3 1 1
Procedure for Finding Graphical Limits
Point: Find the y coordinate of the point that you are approaching from that direction.
Horizontal Asymptote: Find the y coordinate of the equation of the line
Vertical Asymptote: One sided +/- infinity, Double sided does note exist.
0
sinlim
x
x
x
Problem 1:
From both sides you are approaching the point (1,0). The limit is the y coordinate so
-3π -5π/2 -2π -3π/2 -π -π/2 π/2 π 3π/2 2π 5π/2 3π
1
x
y
Solution
0
sinlim 1
x
x
x
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Finding Limits Graphically
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
123456789
x
y
Problem 2:Use the graph to find the following
4a) lim ( )
xg x
4b) lim ( )
xg x
4c) lim ( )
xg x
0d) lim ( )
xg x
0e) lim ( )
xg x
2g) lim ( )
xg x
2h) lim ( )
xg x
2i) lim ( )
xg x
0f) lim ( )
xg x
( )y g x
2
8
DNE
4
4
4
2
2
DNE
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f(2) =
lim ( )x
f x
2lim ( )
xf x
4
lim ( )x
f x
2
lim ( )x
f x
2
lim ( )x
f x
4
lim ( )x f x 4
f(4) =
Problem 3
dne
4
4
4
und
1
2
2
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Finding Finite Limits Numerically Problem 1:Find the following numerically using the table of values x f (x)
4 5.001
3 5.0001
2 34
1 5.0001
0 5.001
-1 5.01
2a) lim ( )
xf x
2b) lim ( )
xf x
2c) lim ( )
xf x
d) (2)f
5
5
5
34
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Problem 2:Find the following limits Numerically
x f (x)
2.02 -9.99
2.01 -9.999
2 Error
1.99 2.9999
1.98 2.999
1.97 2.99
2a) lim ( )
xf x
2b) lim ( )
xf x
2c) lim ( )
xf x
d) 2f
3
10
DNE
Und
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Problem 3:Find the limit
2a) lim ( )
xf x
2b) lim ( )
xf x
2c) lim ( )
xf x
d) 2f
2 , if 2
21 if 2
1 1 if 22
x x
f x x
x x
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Problem 4:Find the limit
0a) lim ( )
xf x
0b) lim ( )
xf x
0c) lim ( )
xf x
d) 0f
23 1 if 0
4 if 0
x xf x
x x
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