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**Limits 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**

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

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**3) Use your calculator to evaluate the limits**

Answer : 16 Answer : no limit Answer : no limit Answer : 1/2

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**One-Sided Limit One-Sided Limits**

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. L a

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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. M a

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**Examples of One-Sided Limit**

1. Given Find Find

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More Examples Find the limits:

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**This theorem is used to show a limit does not exist.**

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

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Limit Theorems

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**Examples Using Limit Rule**

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More Examples

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

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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.**

Divide 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 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: f(a) a

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

f(a) 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 not defined or Exist f(a) is defined to be a number other than the value of limit.

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2- Jump discontinuity

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**3- Infinite discontinuity**

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

Removable discontinuity

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

Jump discontinuity

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**Example Solution X=-3 Removable discontinuity X=0**

Infinite discontinuity X=5 Jump discontinuity X=2 The function is continuous

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

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**Examples Continuous everywhere except at**

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

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**F is continuous everywhere else h is continuous everywhere else**

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

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**Continuous Functions If f and g are continuous at x = a, then**

A polynomial function y = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain.

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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. f (b) f (c) = L f (a) a c b

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

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