MATH 2263: Multivariable Calculus
Determining the existence of a limit of multiple variables
Bruno Poggi
Department of Mathematics, University of Minnesota
September 25, 2016
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Introduction
This document discusses the existence of limits of multiple variables. It is intended as a
supplement to the material covered in Section 14.2 of the book Calculus, Early Trascendentals, 8th Edition, by James Stewart. Probably all the lessons herein contained are
given in the book; our intent is to emphasize certain intuitive notions.
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Is trying limits along some paths enough to determine
the positive existence of a limit?
The answer is an unequivocal no. It may be tempting to conclude that a limit exists
after trying a few different paths and seeing that, along these paths, the limit is the
same number. However, a limit may fail to exist even when one considers infinitely
many paths. Consider the function
f (x, y) =
x19
,
x2 + y 4
whose domain is the whole plane except for the origin, and suppose we want to ascertain
whether
lim f (x, y)
(x,y)→(0,0)
exists. We will see that on uncountably many paths approaching the origin, the function
approaches 0. To see this, let r be any positive real number, and let us approach the
origin (0, 0) along the path
y = xr ,
x → 0+ .
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After a quick algebraic manipulation we see that
limr
y=x
x→0+
x19
1
= lim x19−4r 2−4r
,
2
4
x +y
x
+1
x→0+
whence if 19 6= 4r, the above limit is 0 (why? hint: split up into the cases 19 < 4r and
19 > 4r). In particular, since 19/4 is not a natural number, it follows that for every
polynomial p(x) the limit of f (x, y) along the path
y = p(x), x → 0+
is 0. So, even if you were the enthusiastic computer-saavy student who wrote a program
to determine the limit over millions of polynomials, you would see the limit along these
paths to be 0, and so you might think this gives you evidence that the limit exists and
is 0. However, set r = 19/4, and calculate the limit to see that it is 1. So, the limit
does not exist.
Therefore, a point that must be emphasized is that, while trying limits along a few
different paths can help establish that the limit doesn’t exist, it can never show that the
limit does exist.
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But then, how can it be known that the limit does exist?
One way is if you know that the function is continuous. If the function f (x, y) is
continuous at the point (x0 , y0 ), then the limit
lim
f (x, y)
(x,y)→(x0 ,y0 )
exists and equals f (x0 , y0 ). Recall that morally speaking, limits ”go through” continuous
functions. Polynomials in two variables are continuous at all points in the plane, for
instance, and recall that the composition, addition, and multiplication of continuous
functions preserve continuity. In this way, the existence of a limit can be known if the
function is known to be continuous, so that a calculation is properly justified. As a
quick example, say we wish to calculate
√ 7
3
lim cos πex y+ 6y
(x,y)→(0,0)
√
if the limit exists. Since x3 y + 6y 7 is a polynomial in two variables, it is continuous
on the whole plane, and in particular at the point (0, 0). The one-variable function ez
is continuous, and so is cos(z). Therefore,
!
√
√
lim
x3 y+ 6y 7
x3 y+ 6y 7
(x,y)→(0,0)
lim cos πe
= cos πe
= −1.
(x,y)→(0,0)
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Of course, the problem with the previous method is that there are many functions for
which the continuity is not clear, and establishing the continuity of the function finally
comes down to showing that the limit exists anyway. In this case we have to finally
resort to the definition of the limit, which involves the much-dreaded epsilon,delta’s.
Here, we won’t attempt at a rigorous assessment of the definition of the limit, but we
wish to give an intuitive idea.
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Intuition behind the , δ description of the limit
For simplicity, let us say that the continuous function f (x, y), defined on the whole real
plane, takes the value 0 at (0, 0). At this point, you know that for the limit
lim
f (x, y)
(x,y)→(0,0)
to exist, it must be true that the limit exists (and equals 0, in this case) over every
possible path that approaches the origin. Now, what do all possible paths that approach
the origin have in common? They must all eventually be really close to the origin. So,
if we drew a square in the plane with the origin as the center (call it S), then all paths
approaching the origin must eventually ”be” inside the square, no matter how large or
small we make the square. Now look at the graph of the function f over the square S in
three-dimensional space, and consider the smallest possible box B, extruded from the
square, which contains the whole graph of f over S. See the sketch below for reference.
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Figure 1: Sketch of the description above.
Once you have the above picture in your mind, observe that the limit exists if and
only if, as we decide to make the square S vanishingly smaller, the box B correspondingly becomes vanishingly smaller too. If the continuous function takes a value a 6= 0 at
the origin, simply ”move” the box B so that it is centered at (0, 0, a).
To see that the above model describes limits well, notice what happens when a limit
does not exist: If the (bounded) function f does not have a limit at the origin, then on
at least two paths the function approaches different values c, d. It is clear then that the
height of the box B cannot be made smaller than |c − d| no matter how small we take
the square S to be.
Tinker with the above mental model a little bit to see that the above description
really captures limits in general. How do epsilon and delta relate to this description?
Put simply, one can think of δ as half the side of the square S, and ε as half the height
of the box B (if you used the actual definition of the limit, we would need to use a circle
instead of the square S in the xy−plane, and in this case δ would be the radius of the
circle). Once you feel comfortable with the mental model given here, try to see how
this description helps you understand the epsilon-delta definition of the limit!
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