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.
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