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Limits Graphically & Limit Properties
Day 1 Limits Graphically & Limit Properties
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What is a limit? A limit describes how the output values of a function behave as input values approach some given #, “c” Notation: Read as “limit of f(x) as x approaches c is equal to L”
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3 Kinds of limits THE Limit (double-sided limit) Left-hand limit
Limit of f(x) as x approaches c from either direction. Only exists if left-hand and right-hand limits are the same. Left-hand limit Limit of f(x) as x approaches c from the left side. Right-hand limit Limit of f(x) as x approaches c from the right side.
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Common Misconception #1
A function does not have to be defined at “c” in order for the limit to exist.
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Common Misconception #2
If a function is defined at “c”, f(c) does not necessarily have to equal L.
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Practice
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Practice
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Practice
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Practice 10
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Draw a graph such that
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Draw a graph such that
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Draw a graph such that
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Draw a graph such that
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Draw a graph such that
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Draw a graph such that
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Two Cases for When the Limit is D.N.E. (Does Not Exist)
Behavior differs from the left and right Oscillating Behavior Ex/
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Increase or decrease without bound
Unbounded Behavior Increase or decrease without bound Can you think of a parent function that would fall in this category? More descriptive than DNE
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Some Basic Limits The limit of a constant function is the constant.
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The limit at any x-value on the line y=x IS the x-value itself.
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One more: The limit at any x-value of a function f(x)=xn (where n is an integer) is the x-value raised to the nth power.
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Practice:
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Theorem: Properties of limits
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Theorem: Properties of limit
The limit of a sum is equal to the sum of the limits The limit of a difference is equal to the difference of the limits
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Theorem: Properties of limit
The limit of a product is equal to the product of the limits
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Theorem: Properties of limit
The limit of a quotient is equal to the quotient of the limits
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Theorem: Properties of limit
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