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Limits (Functions of Two Variables)
Department of Mathematics, Sinclair Community College, Dayton,
OH 1
One of the main differences between limits of functions of one
variable and limits of functions of two
variables is that limits of functions of one variable are
considered in an interval on the number line,
whereas limits of functions of two variables are considered in
an open disc in the xy-plane. That is, with
a function of one variable,
| | 0, thereexists a > 0 such that | , | < whenever 0 <
+ < .
Example: Find the limit.
lim,,5 +
Notice that the point (1, 2) does not cause division by zero or
other domain issues. So,
lim,,5 + =
5121 + 2 =
105 = 2
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7/28/2019 Limits of Functions of Two Variables
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Department of Mathematics, Sinclair Community College, Dayton,
OH 2
Example: Find the limit.
lim,,
+
Let x= 0:
= = 0
Now let y= 0:
=
= 1Since we got two different results, the limit does not
exist.
Example: Find the limit.
lim,, 2 +
Let x= 0: = 2 Now let y= 0: = 1
Again, the limit does not exist.
Example: Find the limit.
lim,, +
Let , 0,0 along the line = . Then =
= . This shows that thelimit depends on the choice ofm.
Therefore, the limit does not exist.
When we use the definition of a limit to show that a particular
limit exists, we usually employ certain
key or basic inequalities such as:
| | < + | | < +
+ 1 < 1
+ < 1| | = +
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Department of Mathematics, Sinclair Community College, Dayton,
OH 3
Example: Find the limit.
lim,,
+
Let x= 0:
= 0
Let y= 0:
= 0We suspect that the limit might be zero. Lets try the
definition with L = 0.
| , | < whenever 0 < + <
| , 0| = + = | |
+
Now, since
< 1, then
| |
< | |. So we then have
| | + < | | = < + = 0 + 0 <
Therefore, if = , the definition shows the limit does equal
zero.
Example: Show that the following limit does not exist.
lim,, +
The domain of the function contains all the points on the
xy-plane except for (0, 0). To show
that the limit does not exist, we will approach (0, 0) on two
different paths, the x-axis and = .
Along the x-axis: lim,, 0 + 0
= lim,,1 = 1
Along the line
=:
lim,, +
= lim,, 0
2
= 0
Therefore, since we got different results, the limit does not
exist.
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Department of Mathematics, Sinclair Community College, Dayton,
OH 4
Exercises: Determine whether the following limits exist.
1. lim,, +
2. lim,, 3 +
3. lim,, + 5
2
4. lim,, sin+4
5. lim,,
+
6. lim,,2 +
7. lim,,
+
Answers:
1. No
2. Yes
3.52
4.
5. No
6. No
7. No
