LimitsFunctions of oneand TwoVariables
Limits for Functions of One Variable.What do we mean when we say that
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 LimitaLMeans 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.L+TL-Ta+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!
This isnt True for This function!aLL+TL-TNo 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!
Changing the value of L doesnt help either!
Functions of Two VariablesHow does this extend to functions of two variables? We can start with informal language as before:
means that as (x,y) gets closer and closer to (a,b) , f(x,y) gets closer and closer to L.
Closer and CloserThe words closer and closer obviously have to do with measuring distance. In the real numbers, one number is close to another if it is within a certain tolerance---say no bigger than a+.01 and no smaller than a-.01.In the plane, one point is close to another if it is within a certain fixed distance---a radius!r
What about those strips?The vertical strip becomes a cylinder!
Horizontal Strip?LL+TL-TThe 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+TL lies on the z-axis. We are interested in function values that lie between z=L-T and z=L+T
Putting it All TogetherThe part of the graph that lies above the green circle must also lie between the two horizontal planes.
Defining the LimitMeans 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.Once again, the pt. ((a,b), f(a,b)) can be anywhere (or nowhere) !