LimitsLimits
Functions of oneFunctions of one
and Twoand Two
VariablesVariables
Limits for Functions of One Limits for Functions of One Variable.Variable.
What do we mean when we say thatWhat do we mean when we say that
lim ( ) ?x a
f x L
Informally, we might say that as x gets “closer and closer” to a, f(x) should get “closer and closer” to L.
This informal explanation served pretty well in beginning calculus, but in order to extend the idea to functions of several variables, we have to be a bit more precise.
Defining the LimitDefining the Limit
a
L
Means that •given any tolerance T for L• we can find a tolerance t for a
such that •if x is between a-t and a+t, but x is not a,• f(x) will be between L-T and L+T.
lim ( )x a
f x L
L+T
L-T
a+ta-t
(Graphically, this means that the part of the graph that lies in the yellow vertical strip---that is, those values that come from (a-t,a+t)--- will also lie in the orange horizontal strip.)
Remember: the pt. (a,f(a)) is excluded!
a
L
lim ( )x a
f x L
L+T
L-T
No amount of making the Tolerance around a smaller is going to force the graph of that part of the function within the bright orange strip!
This isn’t True for This This isn’t True for This function!function!
Changing the value of L doesn’t help either!
a
L
In fact, there is no that will help us. lim ( ) does not exist.x a
L f x
L+T
L-T
Functions of Two VariablesFunctions of Two Variables
How does this extend to functions of two variables? How does this extend to functions of two variables?
We can start with informal language as before: We can start with informal language as before:
means that as means that as (x,y)(x,y) gets “closer and closer” to gets “closer and closer” to (a,b)(a,b) , , f(x,y)f(x,y) gets closer and closer to gets closer and closer to LL..
( , ) ( , )lim ( , )
x y a bf x y L
““Closer and Closer”Closer and Closer”
The words “closer and closer” obviously have to do with The words “closer and closer” obviously have to do with measuring distance. measuring distance.
In the real numbers, one number is “close” to another if it is In the real numbers, one number is “close” to another if it is within a certain tolerance---say no bigger than within a certain tolerance---say no bigger than a+.01a+.01 and no and no smaller than smaller than a-.01.a-.01.
In the plane, one point is “close” to another if it is within a In the plane, one point is “close” to another if it is within a certain fixed distance---a radius!certain fixed distance---a radius!
(a,b)
r(x,y)
What about those strips?What about those strips?
(a,b)r
(x,y)
The vertical strip becomes a cylinder!
Horizontal Strip?Horizontal Strip?
L
L+T
L-T
The horizontal strip becomes a “sandwich”!
Remember that the function values are back in the real numbers, so “closeness” is once again measured in terms of “tolerance.”
The set of all z-values that lie between L-T and L+T, are “trapped” between the two horizontal planes z=L-T and z=L+T
L lies on the z-axis. We are interested in function values that lie between z=L-T and z=L+T
Putting it All TogetherPutting it All Together
The part of the graph that lies above the green circle must also lie between the two horizontal planes.
if given any pair of horizontal planes about L, we can find a circle centered at (a,b) so that the part of the graph of f within the cylinder is also between the planes.
( , ) ( , )lim ( , )
x y a bf x y L
Defining the LimitDefining the Limit
Means that •given any tolerance T for L• we can find a radius r about (a,b)
such that •if (x,y) lies within a distance r from (a,b), with (x,y) different from (a,b) ,• f(x,y) will be between L-T and L+T.
( , ) ( , )lim ( , )
x y a bf x y L
L+T
L-T
L
Once again, the pt. ((a,b), f(a,b)) can be anywhere (or nowhere) !
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