Limit of a Function

Section 1 5 .
-

The Limit of a Function 08/26/2022

Math 224


X -
1

Consider the function f(x) =
X2-1

The domain of is all X except X= 1 or X = -1


Interval Notation : (-c , -1)u(-1 1) v(1 ,
c)
,




Let's examine the behavior of f when X is near 1
.




Y
XI X

X f(x) X f(x)
X 1
5 S 400
-




f(x) = 667 1
I
. .
.




X3 -
1
9 S26 1 1 476
2
· .
.
j
·




X
. 99 · So2 1 01
.
. 497 lim =
X >
I

999 ·
500 1 001
.
· 499

19999 . Soo 1 0001
. · 499




"Intuitive"Definition of a Limit :
Suppose f(x) is defined when X is near a number a.


limf(x) L if of
Then we write
X >a
=
we can make the values f(x)
arbitrarily close to L
by
restricting X to be sufficiently close to a Con either side of al but not
equal to a.



Y Y Y



L A
L L

·
(a f(a)
,




X X X
A A A




limf(x) = L lim f(x) = L lim f(x) = L
X U Xa Xa




One-Sided Limits

consider the function H(t) = [8
Y



Lim (t) 0
=



y = H(t)
I


(t)
[m H = 0


-t


LimH(t) = Do Not Exist because
LimH(t) limit


We write limf(x) if f(x)
X > G-
= - we can make the values
arbitrarily close to 1
by
taking X to be
sufficiently close to a with X less than a.

We write limf(x) L =

if we can make the values f(x) arbitrarily close to 1
by
Xa+

taking X to be
sufficiently close to a with X
greater than a.