Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity

Limit L a

Limits, Graphs, and Calculators

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

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

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

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

Examples of One-Sided Limit 1. Given Find Find

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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

Limit Theorems

Examples Using Limit Rule

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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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The Squeezing Theorem See Graph

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

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Infinite Limits For all n > 0, More Graphs

Examples Find the limits

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

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

Examples

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

Asymptotes

Examples Find the asymptotes of the graphs of the functions

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

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

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

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.

2- Jump discontinuity

3- Infinite discontinuity

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

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

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

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

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

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

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.

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

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.