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Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity

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Limit a L

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Limits, Graphs, and Calculators Graph 1 Graph 2

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

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Answer : 16 Answer : no limit Answer : 1/2 3) Use your calculator to evaluate the limits

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The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L One-Sided Limit One-Sided Limits

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The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. a M

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1. Given Find Examples Examples of One-Sided Limit

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Find the limits: More Examples

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For the function Bu t This theorem is used to show a limit does not exist. A Theorem

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Limit Theorems

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Examples Using Limit Rule Ex.

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More Examples

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Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Notice form Factor and cancel common factors Indeterminate Forms

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More Examples

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The Squeezing Theorem See Graph

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Limits at Infinity For all n > 0, provided that is defined. Ex. Divide by

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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 limits

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Limit and Trig Functions From the graph of trigs functions we conclude that they are continuous everywhere

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Tangent and Secant Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers

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Examples

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Limit and Exponential Functions The above graph confirm that exponential functions are continuous everywhere.

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Asymptotes

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Examples Find the asymptotes of the graphs of the functions

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Continuity A function f is continuous at the point x = a if the following are true: a f(a)f(a)

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a f(a)f(a)

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Definition of continuity The function f is continuous at the number c if Defined at c

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Types of Discontinuity 1- Removable discontinuity f(a) is defined to be a number other than the value of limit. f(a) is not defined or Exist

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2- Jump discontinuity

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3- Infinite discontinuity

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Discuss the continuity of f(x) at x = 2, where Removable discontinuity Example Solution

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Discuss the continuity of f(x) at x = 0, where Jump discontinuity Example Solution

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X=-3 Removable discontinuity X=0 Infinite discontinuity X=5 Jump discontinuity X=2 The function is continuous Example Solution

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What value should b assigned to make the function continuous at x = 1 Example Solution

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At which value(s) of x is the given function discontinuous? Continuous everywhere Continuous everywhere except at Example s

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and Thus h is not cont. at x=1. h is continuous everywhere else and Thus F is not cont. at F is continuous everywhere else

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Continuous Functions A polynomial function y = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain. If f and g are continuous at x = a, then

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Intermediate Value Theorem If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. ab f (a) f (b) L c f (c) =

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

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