limits
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Limits
“nearness”
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Consider a polygon inscribed in a circle
The Idea of Limits
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n=3 n=4 n=5 n=6 n=7 n=8‘As number of sides of polygon increases, its area
approximates the area of the circle’‘limit of Area of polygon is the Area of the circle’
As n approaches infinity , Lim Area of polygon = Area of the circle
The Idea of Limits
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Consider the function:
The Idea of Limits 2)( xxg
2)( xxg
x
y
O
2
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
g(x) 3.9 3.99 3.999 3.9999 3.0001 4.001 4.01 4.14
4)(lim2
xgx
As x approaches to positive 2 at both directions
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Fundamental Rules of Limits.1. The Constant Rule
– When we take the limit of a constant, non-changing function, the limit will simply be that constant.
2. The Sum Rule– If two sequences have limits that exist, then the limit of
the sum of sequences is the sum of the limits of the sequences.
3. The Multiplication Rule– If two sequences have limits that exist, then the limit of
the product is the product of the limits.
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Fundamental Rules of Limits.
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Techniques in calculating Limits
T1: Limits By Direct Substitution
T2: Limits by Factoring
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Type 3a: Limits by Rationalization
Techniques in calculating Limits
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T3b: Limits by Rationalization
Techniques in calculating Limits
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T4a: Limits at Infinity
Techniques in calculating Limits
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T4b: Limits at Infinity
Techniques in calculating Limits
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T5: Trigonometric Limits
Techniques in calculating Limits
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T6: Limits Involving Number e
Techniques in calculating Limits
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Try me!!
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Try me!!
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1)(lim0
xhx
1)(lim0
xhx
)(lim0
xhx does not
exist.
Two Sided limit
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