CALCULUS
F O R M AT I V E A S S E S S M E N T
Write your name and date on the provided sheet of paper. Answer the following questions and submit the
paper by the end of the lesson. You may answer the questions as we go with the discussions.
1) Using your own words/understanding, what is a limit?
2) Give an example of a scenario where limit exists.
3) When you are given a function, how do you find the limit as the variable approaches ∞?
4) When you are given a function, how do you find the limit as the variable approaches a certain number?
5) Name all situations when the limit does not exist.
INTRODUCTION TO
LIMITS
What is a limit?
Let’s discuss the derivation of the
area of a circle
(and circumference)
A GEOMETRIC EXAMPLE
• Look at a polygon inscribed in a circle
What do you notice as the number of sides of
the polygon increases?
A GEOMETRIC EXAMPLE
• Look at a polygon inscribed in a circle
As the number of sides of the polygon
increases, the polygon is getting closer to
becoming a circle.
• http://www.mathopenref.com/circleareaderive.html
If we refer to the polygon as an n-gon,
where n is the number of sides
we can make some mathematical statements:
• As n gets larger, the n-gon gets closer to being a circle
• As n approaches infinity, the n-gon approaches the circle
• The limit of the n-gon, as n goes to infinity is the circle
The symbolic statement is:
lim(n - gon) = circle
n ®¥
The n-gon never really gets to be the circle, but
it gets close - really, really close, and for all
practical purposes, it may as well be the circle.
That is what limits are all about!
FYI
Archimedes used this method WAY before
calculus to find the area of a circle.
AN INFORMAL DESCRIPTION
If f(x) becomes arbitrarily close to a single number
L as x approaches c from either side, the limit
for f(x) as x approaches c, is L. This limit is
written as
lim f ( x) = L
x ®c
Numerical
Examples
NUMERICAL EXAMPLE 1
Let’s look at a sequence whose nth term is given by:
n
n +1
What will the sequence look like?
½ , 2/3, ¾, 4/5, ….99/100,...99999/100000…
W H AT I S H A P P E N I N G T O T H E T E R M S O F T H E S E Q U E N C E ?
½ , 2/3, ¾, 4/5, ….99/100,….99999/100000…
Will they ever get to 1?
n
lim =1
n ®¥ n + 1
Numerical Example 2
Let’s look at the sequence whose nth term is given by
!
"
1, ½, 1/3, ¼, …..1/10000,....1/10000000000000…
As n is getting bigger, what are these terms approaching?
1
lim = 0
n ®¥ n
Graphical
Examples
GRAPHICAL EXAMPLE 1
1
f ( x) =
x
As x gets really, really big, what is happening to the height, f(x)?
1
lim = 0
x ®¥ x
As x gets really, really small, what is
happening to the height, f(x)?
Does the height, or f(x) ever get to 0?
1
lim = 0
x ®-¥ x
GRAPHICAL EXAMPLE 2
f ( x) = x 3
As x gets really, really close to 2, what is happening to the height, f(x)?
lim x = 8 3
x®2
GRAPHICAL EXAMPLE3
-7
-4
Find lim f ( x) lim f ( x) = -4
x ®- 7 x ® -7
not 6!
GRAPHICAL EXAMPLE4
Use your graphing calculator to graph the following:
ln x - ln 2
f ( x) =
x-2
ln x - ln 2
GRAPHICAL EXAMPLE4
f ( x) =
x-2
Find lim f ( x)
x®2
TRACE: what is it approaching?
TABLE:
Set table to start at 1.997 with
increments of .001 (TBLSET)
As x gets closer and closer to 2, what is the value of f(x) getting closer to?
Does the value of f(x) exist when x = 2?
ln x - ln 2
f ( x) =
x-2
lim f ( x) ZOOM Decimal
x®2
lim f ( x) = 0.5
x®2
Limits that
Fail to Exist
Nonexistence Example 1: Behavior that
Differs from the Right and Left
What happens as x
approaches zero?
The limit as x approaches zero does not exist.
1
lim = does not exist
x ®0 x
NONEXISTENCE EXAMPLE 2:
UNBOUNDED BEHAVIOR
Discuss the existence of the limit
1
lim 2
x ®0 x
1
lim = does not exist
x ®0 x
NONEXISTENCE EXAMPLE 3:
OSCILLATING BEHAVIOR
Discuss the existence of the limit
1
lim sin
x ®0 x
Put this into your calc
set table to start at -.003 with
increments of .001
X 2/π 2/3π 2/5π 2/7π 2/9π 2/11π X 0
Sin(1/x) 1 -1 1 -1 1 -1 Limit does
not exist
COMMON TYPES OF
B E H AV I O R A S S O C I AT E D W I T H
NONEXISTENCE OF A LIMIT
WHEN CAN I USE SUBSTITUTION
TO FIND THE LIMIT?
• When you have a polynomial or rational function with nonzero denominators
