Limits in Calculus
JON PIPERATO
CALCULUS
GRADES 11-12
Limits in Calculus, an Introduction

Limits are a major component to Calculus, which is used everyday

Integrals take the limit and use it to find an area under a curve

Finding surface area, volume, distances etc.

Electric charge

All engineering and all science
Navigation Through the Unit

All buttons such as these will take you to the next or previous slide

Any information button will take you to the beginning of the lesson
that is specified

The home button will bring you back to the main menu

The question mark will take you to the review problems from that
lesson

Reveals answers to example problems
Main Menu

Lesson 1: Intuitive Definition and Graphing

Lesson 2: Limits in Tables

Lesson 3: Limits that fail to exist

Lesson 4: Algebraic Limits

Lesson 5: Piecewise Functions

Lesson 6: Infinite Limits

Lesson 7: Discontinuity
**Clicking the information button will take you to the beginning of
each lesson
**? Button takes you to review problems
Lesson 1:
Intuitive Definition & Graphing

A limit of a function is described as the behavior of that function as it
approaches a specific point

EX:
 This reads: as x approaches c, the function f(x)
approaches the real number L.

In other words, as x gets closer to c, that f(x) gets closer to L…THIS
DOES NOT MEAN f(c)=L
Limits in Graphing
lim f(x)= 2
lim f(x)= DNE
lim f(x)= 4
lim f(x)= -1
X
x
4
-1
X
X
-4
2-
lim f(x)= -2
lim f(x)= -1
lim f(x)= 3
f(2)= 3
X
X
-4+
-4-
X
2+
Limits in Graphing Continued

For limits approaching an xvalue with two points, if a
direction is not specified, the
limit DNE
Ex. lim f(x)= DNE
X

If a direction is specified, use you finger
and follow the line from that direction
until you reach the point.
Ex. lim f(x)= -2
X

-4
-4+
If a specific function is given, you are
looking for the point that is not connect
to the lines in the graph
Ex. f(2) = 3
Examples of Intuitive Definition

Write the following limits in sentence form:

1)

2)

3)
Click here to
see answers!
Answers to Intuitive Definition

1)
the limit of f(x) as x approaches a is L

2)
the limit of 1/x as x approaches infinity is 0

3)
the limit as 1/x as x approaches 0 is infinity
Limit Graphing Example
lim f(x)=
X
-4
lim f(x)=
X
-2
X
1
X
1-
lim f(x)=
lim f(x)=
lim f(x)=
X
Click here to
see the
answers!
1+
f(1)=
f(-2)=
Graphing Limits Answers
lim f(x)= -2
X
-4
lim f(x)= 3
X
-2
X
1
X
1-
lim f(x)= DNE
lim f(x)= -3
lim f(x)= 4
X
1+
f(1)= 2
f(-2)= 5
You Have Completed Lesson 1!
Summary

Limits are defined as the characteristics of the function as it
approaches a specific point.

Intuitive Definition is writing a limit in sentence form

Graphing limits can be used to find a specific location of functions
as they approach a specific point on a graph
Lesson 2:
Limits in Tables

There are multiple ways to solve for limits in Calculus

This is a second option to solving limits

For this lesson, you will need a TI-84 calculator to complete the tasks
Limits in Tables

The first step of solving limits through tables is to identify what
variable x is going to in the limit
lim f(x2-9/x+3)=
x
-3
Next, knowing that x is approaching -3, you must construct your table
using numbers close to the variable in the statement above.

Numbers such as -3.1, 3.01, 3.001 & -2.9, -2.99, -2.999

Then put the equation into the calculator

Finally use the calculator to find the missing values and estimate
what the pattern of the table is

Don’t panic, we will go through problems together
Limits in Tables

Here is an example of what the table from the previous problem
would look like
-3.1
-6.1
-3.01
-6.01
-3.001
-6.001
-3
-2.999
-5.999
-2.99
-5.99
-2.9
-5.9
Once the numbers are
found, you must use the
pattern to figure out was 3 is equal to…it is equal to
-6
Therefore lim f(x)= -6
x
-3
Limit in Tables Examples

Try these limits with a partner:
Click here
to see the
answers!
Limits in Tables Answers
= 40
= 1/2
=6
You Have Completed Lesson 2
Summary

In conclusion we discovered a way to solve for a limit using our
calculators and tables

Without the proper calculator this problem will be a pain, so please
see me if you must borrow one!
Lesson 3:
Limits That Fail to Exist

A limit doesn't exist if the function is not continuous at that point.

To check if a limit exists or not, graphically, you must approach it
from the left and right side and if they are not equal, they do not
exist.

One type of limit that fails to exist is a jump.


Another is a vertical asymptote


A jump can be found in the graphing of the function |x|/ x
A vertical asymptote is when x=0
Lastly, f(x) oscillates between two fixed values as x approaches c
Examples Of Discontinuities

Match the following discontinuities with the possible graph
Sin (1/x)
|x|/ x
Click here to
see the
answers!
1/x2
Answer to Discontinuities
Sin (1/x)
|x|/ x
1/x2
You Have Completed the Lesson!
Lesson 4:
Algebraic Limits

We can solve certain limits with our knowledge of algebra!

All we have to do is plug the number x is going towards in the
equation


But of course there is a catch!
You cannot just simply plug that number in if it makes the equation
false.

We cannot make the denominator zero
Algebraic Limits

An example of the information on the previous slide is:

Since plugging -2 in will give us zero in the denominator, we must do
some solving…with algebra!!!!

However, if it plugs in without a problem, then you just solve!
Algebraic Limits

There are three ways to solve for a limit

1) Limits by direct substitution

2) Limits by Factoring…Yay!

3) Limits by Rationalization

A video on the next slide will explain how to solve the
following limits
Algebraic Limits Video
Algebraic Limits


Limits by direct substitution

Exactly how it sounds

Just plug the number in!
Limits through factoring


Use factoring to cross out unwanted denominator
Limits through Rationalization

Use the reciprocal to get rid of the unwanted denominator
Algebraic Limit

Example:

Select one to be your answer:

A) 10
B) √5
C) 2
D) ± 2
You Are Incorrect, Please try again
Refer to previous slide if necessary, or raise your
hand for assistance
You are correct! Please return to
menu!
Algebraic Limits

Get with a partner, and solve the following limits algebraically
through factoring

1)

2)

3)

The button for the answers will be on the next page
Algebraic Limits

Stay with your partner and work on the following rationalization
problems

1)

2)
Click here
to see the
answers!
Algebraic Limits Answers
=2
= 1/2
=4
= 11/4
= 1/4
You Have Finished Lesson 4:
Algebraic Limits Summary


In summary there are three different ways to solve for an algebraic
limit

Direct substitution

Factoring

Rationalization
Watch the video before coming to class to get a better
understanding of the lesson!
Lesson 5
Piecewise Functions

A piecewise function look like this :

We already have the skills from previous sections to solve this
problem so do not let looks deceive you!

It is called piecewise because it is broken into pieces on a graph,
but it is no different then solving ordinary functions!
Limits of a Piecewise Function

Example:

Solve for f(5)

Use the function that satisfies the number five

Clearly, you must use the second since 5>0

You then just you substitution from last section

(5)2 = 25
Limits of a Piecewise Function

Let’s try one with a limit

lim f(x) =
x





-2-
So first we must determine which piece to use
Then use direct substitution
We can determine that we must use the first piece since x is less than
negative 2 (coming from the left)
Using direct substitution we plug in -2…2(-2)+8= -4+8= 4
Therefore lim f(x) = 4
x
-2-
Limits of a Piecewise Function
Review

Time for you to try some on your own!

lim f(x) =
x

5
Will the answer be

A) 10

B)5

C) 0
You Are Incorrect, Please try again
Refer to previous slide if necessary, or raise your
hand for assistance
You are correct! Please Proceed!
Limits of a Piecewise Function

One more example just to be sure!

lim f(x) =
x

0
What is your answer?

A) 4

B) 8

C) 5
You Are Incorrect, Please try again
Refer to previous slide if necessary, or raise your
hand for assistance
You are correct! Please return to
main menu!
You Have Finished Lesson 5:
Piecewise Functions Summary

Solving for a piecewise function is easy for us, because we already
learned the process!

The only difference is that the function we are looking at is broken
up into different parts

We will look at why they are broken up in an upcoming lesson

Any questions?
Lesson 6
Infinite Limits

These problems can be extremely simple…as long as you learn the
three rules!!!!!

The first rule: If the powers are the same (of the variable) in the
denominator and numerator, then your answer will be the
coefficient’s of the variables.

The second rule: If the power of the variable in the numerator is
higher then the denominator, then your answer is ∞ (infinite)

The third rule: If the power of the variable in the numerator is less
than the denominator, your answer is 0
Infinite Limits

Refer to this table if you get confused
Infinite Limits

Lets try some examples!

First lets find the variables and what power they are to…

Since they are to the same power, we must you rule number one,
which is to take their coefficient.

So the coefficient of x is one and the coefficient of 3x is 3.

Therefore your answer would be 1/3
Infinite Limits

Example:

Looking at the variables, we can conclude that they have different
powers, and the denominator is bigger.

Therefore we must use rule number three.

With our knowledge we know that rule three makes our answer 0
Infinite Limits

One last example!

We can conclude that the numerators power is greater than the
denominator, which means we will use the second rule.

But wait!!!!! –do not ignore the negative sign in front of the infinity
symbol

In this case the negatives will cancel out giving you a positive ∞
Infinite Limits Examples

Try some of these examples!

1)

2)

3)

4)
Infinite Limit Answers
= 1/3
= 1/2
=0
= -∞
You Have Completed Lesson 6!
Lesson 7
Asymptote Graphs (Discontinuities)

Remember in lesson 5, piecewise functions, I said we would look at
why they were in pieces?

There are three different reasons for asymptotes in a graph that will
be touched on in the lesson

Jumps…Piecewise functions

A limit at infinity (1/x or 1/x2)

A removable discontinuity
Discontinuities
 Jump
Discontinuities
 Limit
at infinity
Discontinuities
 Removable
Lesson 7 Discontinuity


We talked about the three types of discontinuities and examples of
each

Removable

Jumps

Limits at infinity
Which of the three types are scene in the graphing approach of
solving limits?

A) Removable

B)Jumps

C) Limits at Infinity
You Are Incorrect, Please try again
You are correct! Please to the main
menu!
The Lesson is Complete!

Click the button to return to the title slide for the next student!