What's a Limit?
A GEOMETRIC EXAMPLE

Let's look at a
polygon inscribed in a
circle... If we increase
the number of sides
of the polygon, what
can you say about the
polygon with respect
to the circle?
A GEOMETRIC EXAMPLE

As the number of
sides of the polygon
increase, the polygon
is getting closer and
closer to becoming
the circle!
A GEOMETRIC EXAMPLE

If we refer to the polygon as an n-gon, where
n is the number of sides, we can make some
equivalent mathematical statements. (Each
statement will get a bit more technical.)
 As
n gets larger, the n-gon gets closer to being
the 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!
A GEOMETRIC EXAMPLE


“The limit as n goes to infinity of n-gon is a
circle”
The n-gon never really gets to be the circle, but
it will get darn close! So close, in fact, that, for all
practical purposes, it may as well be the circle.
That's what limits are all about!
SOME NUMERICAL EXAMPLES




Example 1: Let's look at the sequence whose
nth term is given by n/(n+1).
Recall, that we let n=1 to get the first term of the
sequence, we let n=2 to get the second term of
the sequence and so on.
What will this sequence look like?
1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,...
99999/100000,...
SOME NUMERICAL EXAMPLES
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
What's happening to the terms of this
sequence?
Can you think of a number that these terms are
getting closer and closer to?
1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,...
99999/100000,...
Yep! The terms are getting closer to 1!
But, will they ever get to 1?
 Nope!

So, we can say that these terms are
approaching 1, therefore, the limit is 1.
SOME NUMERICAL EXAMPLES
As n gets bigger and bigger, n/(n+1) gets
closer and closer to 1...
 Mathematical notation (Calculus language)

SOME NUMERICAL EXAMPLES

An visual of numerical
limits we can think of
is an ASYMPTOTE
 If
we graph f(n) =
n/(n+1)
 As n gets larger and
larger we’ll get the
graph on the right
 The horizontal
asymptote is y = 1
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
SOME NUMERICAL EXAMPLES
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EXAMPLE 2:Now, let's look at the sequence
whose nth term is given by 1/n. What will this
sequence look like?
1/1, 1/2, 1/3, 1/4, 1/5,... 1/10,... 1/1000,...
1/1000000000,...
As n gets bigger, what are these terms
approaching?
That's right! They are approaching 0.
How can we write this in Calculus language?
SOME NUMERICAL EXAMPLES
Sometimes you need to alter the
expression to get the limit
 Example 3: Find the limit of

1  3n 2
 1

lim

lim

3


2
n  n 2
n 
n


1
 lim 2  lim 3
n  n
n 
= 0+3
=3
YOUR TURN

Find the limit

It’s 5/2
2  5n
lim
n 
2n
Limits that do not exist



Some sequences just do not have a limit.
n
For example, lim
n 
The sequence for this limit is
 1,

2, 3 … 999,999,999, 1,000,000,000 … a bazillion
These
 numbers just keep getting bigger and
bigger so there is no limit for this sequence
Limits that do not exist

Another example of a sequence that does not
have a limit is
lim (1)n
n 

The sequence is
 -1,
1, -1, 1 … -1, 1
 This one can’t make up its mind. It just keeps
going back and forth from -1 to 1 so this one
also has no limit.
Your Turn

Find the limit
2
2

5n

4n
 A) lim
n 
2n


(1) n n
 B) lim
n  8n  1