Tropic of Calculus » Multivariable Lesson 2.2: Limits, Again

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Multivariable Lesson 2.2: Limits, Again

Our ultimate goal, of course, is to extend our previous methods of calculus so we can analyze functions of several variables in ways analogous to those we used on functions of a single variable. We will soon address the task of differentiating multivariate functions; first, however, we will agree on what it means to limit a multivariable function.

Recall our original loose definition of the limit $\lim_{x\to a}f(x)=L$ : as $x$ approaches $a$ , the value of $f(x)$ approaches $L$ . Remember that for the limit to exist, $f(x)$ has to approach $L$ no matter which direction $x$ is approaching $a$ from either the left or the right.

Using epsilon-delta, we refined that definition significantly: given any $\epsilon$ , we can find a $\delta$ such that, when $x$ is within $\delta$ of $a$ , we are guaranteed that $f(x)$ is within $\epsilon$ of $L$ . Or, in mathematical language, for any $\epsilon$ , there exists a $\delta$ such that $0<\vert x-a\vert<\delta$ implies that $\vert f(x)-L\vert<\epsilon$ .

It’s actually pretty easy to extend this definition of limits to multivariate functions. For the limit $\lim_{(x,y)\to (a,b)}f(x,y)=L$ : as the input $(x,y)$ approaches $(a,b)$ , the value of $f(x,y)$ approaches $L$ . Note that this must be the case no matter which direction $(x,y)$ is approaching $(a,b)$ from the left, right, or—actually, there are a lot of ways to approach an ordered pair. But we’ll get back to that.

What about epsilon-delta? Given any $\epsilon$ , we can find a $\delta$ such that, when $(x,y)$ is within $\delta$ of $(a,b)$ , we are guaranteed that $f(x,y)$ is within $\epsilon$ of $L$ . But what does it mean for an ordered pair to be within a certain distance from another ordered pair? Well, instead of dealing with one-dimensional distance (subtraction), we will deal with two-dimensional distance (the distance formula). So, more formally, our multivariate limit means that, for any $\epsilon$ , there exists a $\delta$ such that $0<\sqrt{(x-a)^2+(y-b)^2}<\delta$ implies that $\vert f(x,y)-L\vert<\epsilon$ . (The parallels might be clearer if you rewrite our previous definition $0<\vert x-a\vert<\delta$ as the equivalent $0<\sqrt{(x-a)^2}<\delta$ .)

We can visualize the previous definition as restricting our input to a segment with length $2\delta$ (minus $a$ itself), to guarantee our output will be within a segment. In that way, we’re forming a little rectangle that our function must fit into. For functions of two variables, we are restricting our input to a circle with radius $\delta$ (minus $(a,b)$ itself), to guarantee our output will be within a segment. In that way, we’re forming a little cylinder that our function must fit into. [diagram of both!]

So far so good. But while there are only two ways to approach $a$ , there are many more ways to approach $(a,b)$ ! We can break up the limit into the limits of $x$ and $y$ separately, depending on how we choose to approach $(a,b)$ . For example, we can first have $x$ approach $a$ , and then have $y$ approach $b$ . $\lim_{(x,y)\to (0,0)} \sin(x^2+y^2)=\lim_{y\to 0}\left(\lim_{x\to 0}\sin(x^2+y^2)\right)=\lim_{y\to 0}\sin(0+y^2)=0$

We can interpret this as approaching $(0,0)$ along the $y$ -axis, that is, the line $x=0$ . This is because we first restricted ourselves to the line $x=0$ , then we approached $(0,0)$ along that line. [graph]

Remember, we said that for the limit to exist, it must be the same along any path. What about the graph of $f(x,y)=\frac{x^2-y^2}{x^2+y^2}$ ? If we approach along the $x$ -axis, we substitute $y=0$ into the equation and are left with the limit $\lim_{x\to 0} \frac{x^2-0}{x^2+0}$ which evaluates to one. If we approach along the $y$ -axis, on the other hand, we substitute $x=0$ and get the limit $\lim_{y\to 0} \frac{0-y^2}{0+y^2}$ which is of course negative one. Since the limit depends on the path we take, the limit does not exist. If we look at the graph, it becomes clear why the function doesn’t approach the same value from both of these directions. [graph]

Some functions behave the same way along the axes but act differently along other lines like $y=x$ . Yet other functions need to be limited along a path like $y=x^2$ . Remember that it’s very easy to show a limit doesn’t exist—just find two paths that limit differently. Finding these paths isn’t as hard as it seems, because with a little practice you should be able to see which substitutions may yield different limits. But to be honest, you won’t need to do this very much. As long as you accept our revised concept of the limit, that should be enough for you to understand partial derivatives. For class you may need to practice disproving or even proving these limits (using epsilon-delta) but as to us, we shall move on.