LIMITS AND CONTINUITY
An introduction to Limits and how we will
be using them
LIMIT
We say the limit of f (x) as x approaches a is L and write
lim f ( x)  L
x a
y  f ( x)
L
a
EXAMPLES OF LIMITS
f (x) = 3x+10
Given
Find
lim+ f (x)
x®3
lim f (x) = lim 3x+10 = lim 3(3) +10 =19
x®3
x®3
x®3
THE
 - DEFINITION
OF LIMIT
We say lim f ( x)  L if and only if
x a
for every positive number e ,
there exists a positive number d such that
if 0 <| x- a |< d , then | f (x) - L |< e.
L 
L 
y  f ( x)
L
a
a 
a 
ONE-SIDED LIMITS
The right-hand limit of f (x), as x
approaches a, equals L
written: lim f (x) = L
x®a+
when we can make f (x) arbitrarily
close to L by taking x to be sufficiently
close to the right of a.
y  f ( x)
L
a
EXAMPLES
ì x+ 2, if x > 0
Let f (x) = í
î3x -1, if x £ 0.
Find the limits:
a) lim+ f (x) = lim (x+ 2) = 0 + 2 = 2
x®0
b)
x®0+
lim f ( x) = lim- (3x-1) = 3(0) -1= -1
x 0
x®0
c) lim f ( x) = lim- (x+ 2) =1+ 2 = 3
x 1
x®1
d) lim f ( x) = lim(x+ 2) =1+ 2 = 3
x 1
x®1+
A THEOREM
lim f (x) = L iff lim+ f (x) = L and lim- f (x) = L.
x®a
x®a
x®a
We can use this theorem to show a
limit does not exist
For the
function
ì x+ 2, if x > 0
f (x) = í
î3x -1, if x £ 0.
lim f (x) DNE because lim+ f (x) = 2 and lim- f (x) = -1.
x®0
x®0
x®0
HW 2
Problems 2.3, 2.5, 2.7, 2.8, 2.13, 2.18