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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.

Transcripts

Breaking points 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 included 2

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

The Definition of Limit L a

Examples What do we do with the x?

1/2 1 3/2

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

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

Examples of One-Sided Limit Examples 1. Given Find

More Examples Find the cutoff points:

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

Limit Theorems

Examples Using Limit Rule Ex. Ex.

More Examples

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

More Examples

The Squeezing Theorem See Graph

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

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

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

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

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 .

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

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.

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

More Examples

Infinite Limits For all n > 0, More Graphs

Examples Find the points of confinement

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

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

Examples

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

Asymptotes

Examples Find the asymptotes of the charts of the capacities