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1.1 Notes on Numerical Limits
Consider the limit of this function as x  -1
f ( x) 
2x2  x 1
x 1
Let’s make a table of values to understand this graph:
(Since we can’t just plug in -1, start by substituting numbers to the left and right of -1 into the equation)
x  1
f  x
x  1
f  x
-2
-5
0
1
-1.5
-4
-1.5
-2
The limit as x  -1 from the left is ___________
The limit as x  -1 from the right is __________
f ( x) 
So, the limit of
2x2  x 1
x 1
as x -1 is _________
Exercises:
1. Find the limit of
x
f(x)
6.97
2. Find the limit of
x
f(x)
3.97
g ( x)  x  3
6.98
as x  7
6.99
7
7.01
7.02
7.03
4
4.01
4.02
4.03
-0.99
-0.98
-0.97
3.01
3.02
3.03
g ( x)  x  3 as x  4
3.98
3.99
2 x  1, x  1
h( x )  
2, x  1
3. Find the limit of
as x  -1
x
f(x)
-1.03
-1.02
-1.01
-1
2 x  1, x  1
h( x )  
2, x  1
4. Find the limit of
as x  3
x
f(x)
2.97
2.98
2.99
3
k ( x) 
5. Find the limit of
x
f(x)
1.97
1.98
k ( x) 
6. Find the limit of
x
f(x)
-2.03
x2
x 2  4 as x  2
1.99
2
2.01
2.02
2.03
-2
-1.99
-1.98
-1.97
x2
x 2  4 as x  -2
-2.02
-2.01
Limit Notation:
Two-Sided Notation:
“The limit of f (x) as x approaches a is L”
One-Sided Notation:
“The limit of f (x) as x approaches a from the right/left is L”
Homework: 1.1 Numerical Limits Worksheet