Points of confinement and Continuity .

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Limits and Continuity. Definition. Evaluation of Limits. Continuity. Limits Involving Infinity. Limit. L. a. Limits, Graphs, and Calculators. Graph 1. Graph 2. Graph 3. c) Find. 6. Note: f (-2) = 1 is not involved . 2. 3) Use your calculator to evaluate the limits.
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´╗┐Breaking points and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity

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Limit L a

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Limits, Graphs, and Calculators Graph 1 Graph 2

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Graph 3

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c) Find 6 Note: f (- 2) = 1 is not included 2

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3) Use your adding machine to assess the cutoff points Answer : 16 Answer : no restriction Answer : no restriction Answer : 1/2

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The Definition of Limit L a

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Examples What do we do with the x?

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1/2 1 3/2

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One-Sided Limits One-Sided Limit The right-hand point of confinement of f ( x ), as x methodologies a , measures up to L composed: in the event that we can make the worth f (x) self-assertively near L by taking x to be adequately near the privilege of a . L a

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The left-hand farthest point of f ( x ), as x methodologies a , meets M composed: on the off chance that we can make the worth f (x) self-assertively near L by taking x to be adequately near the left of a . M a

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Examples of One-Sided Limit Examples 1. Given Find

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More Examples Find the cutoff points:

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A Theorem This hypothesis is utilized to demonstrate a farthest point does not exist. For the capacity Bu t

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Limit Theorems

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Examples Using Limit Rule Ex. Ex.

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More Examples

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Indeterminate Forms Indeterminate structures happen when substitution in the utmost results in 0/0. In such cases either calculate or defend the expressions. Notice structure Ex. Element and cross out regular elements

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More Examples

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The Squeezing Theorem See Graph

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Continuity A capacity f is nonstop at the point x = an if the accompanying are valid: f ( a ) a

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A capacity f is constant at the point x = an if the accompanying are valid: f ( a ) a

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At which value(s) of x is the given capacity spasmodic? Illustrations Continuous wherever Continuous wherever with the exception of at

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and Thus F is not cont. at Thus h is not cont. at x=1. F is nonstop wherever else h is ceaseless wherever else

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Continuous Functions If f and g are constant at x = a , then A polynomial capacity y = P ( x ) is persistent at each point x. A discerning capacity is persistent at each point x in its area .

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Intermediate Value Theorem If f is a nonstop capacity on a shut interim [ a , b ] and L is any number between f ( a ) and f ( b ), then there is no less than one number c in [ a , b ] to such an extent that f ( c ) = L . f ( b ) f ( c ) = L f ( a ) a c b

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Example f ( x ) is constant (polynomial) and since f ( 1 ) < 0 and f ( 2 ) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] to such an extent that f ( c ) = 0.

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Limits at Infinity For all n > 0, gave that is characterized. Separate by Ex.

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More Examples

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Infinite Limits For all n > 0, More Graphs

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Examples Find the points of confinement

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Limit and Trig Functions From the diagram of trigs capacities we infer that they are persistent all over the place

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Tangent and Secant Tangent and secant are constant wherever in their area, which is the arrangement of every genuine number

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Limit and Exponential Functions The above chart affirm that exponential capacities are consistent all around.

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Examples Find the asymptotes of the charts of the capacities

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