BC 1-2
Limits 2
Name:
Quick Intro - One-Sided Limits
So far we have looked at a limit as x  a or h  0 for example, but we have not clarified
or emphasized how x approaches a or how h approaches 0, meaning that the approach could be
from either the right side or the left side.
x
We consider the function f ( x) 
. What is lim (x) ? We would like to be able to
x0
x
give one, unique answer to a limit. We need some way to define, clarify, and express what
happens to ƒ at 0. We introduce the following notation:
a
We write lim  ( x)  M if and only if  ( x)  M
xa
xa–
left-hand
limit
whenever x  a and x  a .
Similarly, we write lim  ( x)  L if and only if
xa+
right-hand
limit
x a
 ( x )  L whenever x  a and x  a .
From the graph below, we can see that lim
x0
x
x
 1 and lim
x0
x
x
 1 , but lim
x0
exist (DNE) since these one-sided limits do not match, which means that f ( x) 
to one specific number if I is close to 0.
Find each of the following limits:
x2
lim
(1a)
(b)
x2 x  2
BC 1-2
lim
x2
x2
x2
Lim 2.1
(c)
lim
x2
x2
x2
x
x
x
x
does not
is not close
BC 1-2
Limits 2
Name:
Quick Intro - One-Sided Limits
Find each of the following limits:
(2a)
(3)
lim  x 
(c)

(c)
lim  x 
x 1


lim  x
x1
(b)
lim  x
x1
(c)
k(–2)
lim k ( x)
x2
(b)
(d)
Lim 2.2
lim k ( x)
x2
lim k ( x)
x2

lim  x
x1
 x 2  1, x  2
Given k ( x)  
, find:
 x  5, x  2
(a)
BC 1-2
lim  x 
x 1
2x  1, x  1
Given  x  
, find:
 3x  1, x  1
(a)
(4)
(b)
x 1
(5)
Sketch the graph of y 
1
.
x
What happens to y as x  0– ?
What happens to y as x  0+ ?
We write this as follows:
1
1
lim   and lim  
 x
 x
x0
x0
Find each of the following limits.
1
lim
(6a)
(b)
x4 x  4 2

(7a)
(8)
BC 1-2

lim
x4
x4
x2
(b)
lim tan x
(9)
lim
x2
x 2

1
(c)
x  42
Lim 2.3
x4
 x  4
lim 
x2
x2
lim sec x
x 2
lim

1
x  42
(10) Use the graph of y  f ( x) to find each of the following limits.
y  f ( x)

lim  x 
x0

lim  x 
x2

lim  x 
x0

lim  x 
x2

(0) =

(2) =
lim  x 
x0
lim  x 
x2
(11) Use the graphs of y  f ( x) and y  g ( x) to find each of the following limits.
y  g ( x)
lim g  x  
x  2
lim  g  x   
2
x0
BC 1-2
lim g  x  
lim g  x  
x  2
g(2) =
x2
lim    x   g ( x)  
x0
Lim 2.4
lim  g ( x)  
x0