Determining limit values
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Determining Limit Values. Limits give us a language for describing how the outputs ( y values) of a function behave as the inputs ( x values ) approach some particular value . Sometimes we use direct substitution, factoring, rationalizing or expanding to calculate a limit.

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Determining Limit Values

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Determining limit values

Determining Limit Values

  • Limits give us a language for describing how the outputs (y values) of a function behave as the inputs (xvalues) approachsome particular value.

  • Sometimes we use direct substitution, factoring, rationalizing or expanding to calculate a limit.


Definition of a limit

Definition of a Limit

We write limf(x) = L

x→a

and say “the limit of f(x), as xapproachesa, equals L”

if we can make the values off(x) arbitrarily close to L

(as close to Las we like)

by taking xto be sufficiently close to a, but not equal to a.


Determining limit values

Functions Continuous at ‘a’

A function is continuous when its graph is a single

unbroken curve that you could draw without lifting

your pen from the paper.

That is not a formal definition, but it helps you

understand the idea.


Determining limit values

Functions Continuous at ‘a’

Polynomial Functions are Continuousat all x values

(1, 5)

(1 3)

5

3

lim5 =

x→1

5

lim (– 2x + 5) =

x→3

3


Determining limit values

Functions Continuous at ‘a’

lim() =

x→3


Determining limit values

Functions Continuous at ‘a’

0

0

limf(x) =

x→2

(2, 0)


Determining limit values

If direct substitution can be done without an error

we can determine the function value and limit this way.

lim(7 – 2x) =

x→3

7 – 2(3) = 1

7 – 2(3) = 1

42 + 2(4) = 24

lim(x2 + 2x) =

x→4

42 + 2(4) = 24

lim(2x2 – 3x + 4) =

x→2

2(2)2– 3(2) + 4 = 6

2(2)2– 3(2) + 4 = 6


Limits rational expressions

Limits Rational Expressions

If ris a rational function given by

and c is a real number such that q(c) ≠ 0, then


Determining limit values

Let’s look at some examples to clarify this MATH LANGUAGE


Determining limit values

Remember: Division by 0 is undefined.

Rational Functions are continuous at any x value that does not result in division by zero.


Try it

Try it!

By Substitution

The function is continuous at x = 3


Determining limit values

So what is not continuous

(also called discontinuous)?

Look out for holes, jumps or vertical asymptotes.


Determining limit values

Exploring a Nonexistent Limit.

Rational Functions are discontinuous at any x value that DOES result in division by zero.

undefined

There will be a vertical asymptote

when both function and limit

do not exist


Determining limit values

Exploring a Nonexistent Limit.

x = 1

Use a table to show the

limit does not exist.

Find

  • Use a graph to show the

  • limit does not exist.

The y values are getting further

apart the closer x gets to 1

Since f (1) does not exist and the limit as x→1 does not exist

there is a vertical asymptote at x = 1


Finding limits algebraically when the function is undefined

Finding Limits Algebraically when the Function is Undefined

Direct substitution

won’t WORK!!!

DNE

Factor and reduce

There is a vertical asymptote

f(5) does not exist

Substitute – 5 in forx

Undefined!


Finding limits algebraically when the function is undefined1

Finding Limits Algebraically when the Function is Undefined

Direct substitution

won’t WORK!!!

undefined

HOWEVER as x→5


Finding limits algebraically when the function is undefined2

Finding Limits Algebraically when the Function is Undefined

Factor and reduce

f(5) does not exist

There is a “hole” at

Substitute 5 in for x


Lesson 5 worksheet example 1 the function does not exist at a

Lesson #5 WorksheetEXAMPLE 1: The function DOES NOT exist at ‘a’.

-∞

Approaching 0.1666..


Determining limit values

Lesson #5 Worksheet EXAMPLE 1: Continued

The function f(x) does not exist at x= +3 but the limits of f(x) as x approaches are a different matter.

(-3, 0.166666)

x = 3

undefined


Limit of a piecewise function

Limit of a Piecewise Function

The function g has a limit of 2 as x →1

even though g(1) ≠ 2

(1, 2)

(1, 1)

(1, 1)

The function is discontinuous when x = 1.

There is a “hole” at (1, 2).


Limit of a function involving a radical

YES!

Limit of a Function Involving a Radical

Is there a radical rule?

Let n be a positive integer. The following limit is valid

for all c if n is odd, and is valid for c > 0 if n is even.


Determining limit values

n is odd and > 0

.44


Finding limits

Finding Limits

By Graphing

By Substitution

(3, 2)

2

3


Exploring a nonexistent limits

Use a graph of to show that the following limit does not exist.

Exploring a Nonexistent Limits

There is no point on

the graph wherex= – 3

By substitution:

is a non-real number


Remember this

Remember This!

Multiply by its conjugate

3

NOTE: The answer is a rational number


Rationalizing the numerator

Rationalizing The Numerator

Rational number


Find the following limit by rationalizing the numerator

Find the following Limit by Rationalizing The Numerator

Direct substitution won’t WORK!!!

1


Lesson 5 continued example 3

Lesson 5 continued—Example 3

Direct substitution won’t WORK!!!

Multiply Top and Bottom by the conjugate.

1


Lesson 5 worksheet continued example 2 page 2 practice rationalizing

Lesson 5 Worksheet continued Example 2 Page 2 Practice Rationalizing:

1


Example 3 expanding technique

EXAMPLE 3: Expanding Technique

Again substitution won’t WORK.

Find


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