Continuous functions And Continuous functions And limits limits

POP. If you have to lift your POP. If you have to lift your pencil to make the graph then pencil to make the graph then its not continuous.its not continuous.

Continuous ExamplesContinuous Examples

What is a limit?What is a limit?

As x approaches from left

X=X= XX22+6+61 7

1.9 9.61

1.99 9.9601

1.999 9.996001

1.9999 9.9996

1.99999 9.99996

1.999999 9.999996

X=X= XX22+6+63 15

2.1 10.41

2.09 10.3681

2.009 10.03608

2.0009 10.0036

2.00009 10.00036

2.000009 10.00004

As x approaches from right

2

lim

x 2

lim

x

62

lim 2

xx

Can Be Discontinuous and Can Be Discontinuous and have a limithave a limit

Discontinuous with one-sided Discontinuous with one-sided limitslimits

3

lim

x

3

lim

x

How to find the limit of certain How to find the limit of certain functionsfunctions

The following slides will help you find The following slides will help you find the limit of some basic polynomial the limit of some basic polynomial and rational functions.and rational functions.

This will not help you find more This will not help you find more complicated functions. complicated functions.

Worst case scenario take values really Worst case scenario take values really close to the limit and see if there is a close to the limit and see if there is a trend.trend.

Ways to find a limitWays to find a limit

Is the limit going to a Is the limit going to a constant?constant?

Ex. Ex.

Is the limit going to infinity?Is the limit going to infinity?

Ex. Ex.

7xYES

x

7x

YES

Is the function a polynomial function?Is the function a polynomial function?

Is the function a rational function with Is the function a rational function with two polynomial functions?two polynomial functions?

Limit as x goes to a Limit as x goes to a constantconstant

YES

YES

Is the function a polynomial function?Is the function a polynomial function?

Is the function a rational function with Is the function a rational function with two polynomial functions?two polynomial functions?

Limit as x goes to Limit as x goes to ∞∞

YES

YES

Then just plug in the value of Then just plug in the value of x. x. Polynomial functions are Polynomial functions are continuous for all real continuous for all real numbers. numbers.

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Plug in the value of x Plug in the value of x Do you get Do you get ??

Do you get ?Do you get ?0

#

0

0or

#

0 YES

YES

Then the limit goes to zero.Then the limit goes to zero.

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Look at the highest degree Look at the highest degree of each polynomial.of each polynomial.

Is the highest degree in the numerator?

Is the highest degree in the denominator?

Are the degrees the same in both denominator and numerator?

None of the above?

YES

YES

YES

YES

Then the limit goes to Then the limit goes to infinityinfinity

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Then the limit is the fraction Then the limit is the fraction of the leading coefficientsof the leading coefficients

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Then the limit goes to zeroThen the limit goes to zero

The numerator grows slower The numerator grows slower than the denominator.than the denominator.

It acts like 1/It acts like 1/∞∞

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The function needs to be factored, or The function needs to be factored, or simplified, or multiply by the simplified, or multiply by the

conjugate of the denominator. conjugate of the denominator.

Then plug the value of x back in.Then plug the value of x back in.

You may also take values of x that are You may also take values of x that are approaching what the limit is approaching what the limit is

approaching.approaching.end Return

The endThe end• In general finding a limit of a function is In general finding a limit of a function is

finding the value of f(x) as x approaches a finding the value of f(x) as x approaches a value.value.

• The left side and the right side limits must The left side and the right side limits must be the same in order for it to have a have be the same in order for it to have a have limit at the point.limit at the point.

• The function does not have be continuous The function does not have be continuous at the limit value.at the limit value.

• A graph of the function might be helpful to A graph of the function might be helpful to find the limit, but you cant always depend find the limit, but you cant always depend on them.on them.

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