MAT1234 Handout 2.2
One-Sided Limits
The left-hand limit is
when x
approaches 3.
lim f ( x) 
x 3
Independent of f (3) .
Left-Hand Limit
We write lim f ( x)  L and say “the left-hand limit of f ( x) , as x approaches a, equals
xa
L” if we can make the values of f ( x) arbitrarily close to L (as close as we like) by
taking x to be sufficiently close to a and x less than a.
The right-hand limit is
when x
approaches 2.
lim f ( x) 
x  2
Independent of f (2) .
Right-Hand Limit
We write lim f ( x)  L and say “the right-hand limit of f ( x) , as x approaches a,
x a
equals L” if we can make the values of f ( x) arbitrarily close to L (as close as we like)
by taking x to be sufficiently close to a and x greater than a.
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Limit of a Function
lim f ( x)  L if and only if lim f ( x)  L and lim f ( x)  L
x a
xa
x a
Independent of f (a ) .
Example 1
lim f ( x) 
x  2
lim f ( x) 
x  2
lim f ( x) 
x2
Example 2
lim f ( x) 
x  2
lim f ( x) 
x  2
lim f ( x) 
x2
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Infinite Limits
The left-hand limit does not exist.
lim f ( x) 
xa
is not a number.
The left-hand limit DNE.
lim f ( x) 
xa
The right-hand limit DNE.
lim f ( x) 
xa
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The right-hand limit DNE.
lim f ( x) 
xa
The limit DNE.
lim f ( x) 
xa
The limit DNE.
lim f ( x) 
xa
The limit DNE.
lim f ( x ) 
xa
lim f ( x) 
xa
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Classwork
1. For the function f whose graph is given, state the value of the given quantity, if it
exists. If it does not exist, explain why.
(a) lim f ( x) 
lim f ( x) 
x 1
x 1
l i mf x( )
f (1) 
x 1
(b) lim f ( x) 
x 3
l i mf x( )
x 3
(c) lim f ( x) 
lim f ( x) 
x 3
f (3) 
lim f ( x) 
x 2
x 2
l i mf x( )
f (2) 
x 2
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2. Sketch the graph of an example of a function f that satisfies all the given conditions.
lim f ( x)  1 , lim f ( x)  1 , lim f ( x)  0 , lim f ( x)  1 ,
x 0
x 0
x2
x2
f (2)  1 , f (0) is undefined

Make sure you label your axes and other important aspects of the graph.
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