Introduction to Limits
Section 1.2
What is a
limit?
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.
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, 5/6, ….99/100, 99999/100000…
What is happening to the terms of
the sequence?
½ , 2/3, ¾, 5/6, ….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
1
nth term is given by n
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)?
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
As x gets really, really close to 2, what is
happening to the height, f(x)?
lim x  8
3
x2
3
Graphical Example 3
Find
lim f ( x)
x 7
Graphical Example 3
Use your graphing calculator to graph the following:
ln x  ln 2
f ( x) 
x2
f ( x)
Find lim
x2
As x gets closer and closer to 2, what
is the value of f(x) getting closer to?
Does the function
ln x  ln 2
f ( x) 
x2
exist when x = 2?
lim f ( x )
x2
lim f ( x)  0.5
x2
ZOOM Decimal
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
Discuss the existence of the limit lim f ( x)
x 7
Nonexistence Example 3:
Unbounded Behavior
Discuss the existence of
the limit
1
lim 2
x 0 x
Nonexistence Example 4:
Oscillating Behavior
Discuss the existence of the limit
1
lim sin
x 0
x
X
2/π
Sin(1/x) 1
2/3π
2/5π
2/7π
2/9π
2/11π
X
0
-1
1
-1
1
-1
Limit does
not exist
Common Types of Behavior
Associated with Nonexistence of a
Limit