In the following video we go over the definitions of limit and continuity. The definitions are repeated in text form below the video.
For functions of one variable, the (rough) definition of a limit was:
| ${\displaystyle \lim_{x \to a} f(x) = L}$ if $f(x)$ is close to $L$ whenever $x$ is close to (but not equal to) $a$. |
We made that precise by saying exactly what `close to' means. We used the letter $\epsilon$ for how close $f(x)$ has to get to $L$, and $\delta$ for how close $x$ is to $a$:
| Precise definition of limit: ${\displaystyle \lim_{x \to a} f(x) = L}$ if, for every number $\epsilon > 0$ there exists a number $\delta >0$ such that $|f(x) -L| < \epsilon$ whenever $0<|x-a|<\delta$. |
The rough and precise definitions of limits of functions of two (or more) variables work the same way:
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(Rough definition) $\displaystyle{ \lim_{(x,y) \to (a,b)} f(x,y) = L}$
if $f(x,y)$ is close to $L$ whenever the point $(x,y)$ is close to
(but not equal to) $(a,b)$.
(Precise definition) $\displaystyle{ \lim_{(x,y) \to (a,b)} f(x,y) = L}$ if, for every number $\epsilon > 0$ there exists a number $\delta >0$ such that $\displaystyle{|f(x) -L| < \epsilon}$ whenever $\displaystyle{0<\|\langle x-a, y-b \rangle\|<\delta}$. |
The only difference is what we mean by `close to $(a,b)$'. We mean the distance in the plane: $$\|\langle x-a, y-b \rangle\| = \sqrt{(x-a)^2 + (y-b)^2}.$$This is small whenever $x$ is close to $a$ and $y$ is close to $b$. Just one of them being close isn't good enough.
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Example 1: Is $\displaystyle{\lim_{(x,y) \to (2,3)} xy^2
= 18}$?
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Solution: Yes, since a number close to 2 times the square of a number close to 3 is close to $2(3)^2=18$. |
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Example 2: If $$f(x,y) = \frac{x^2-y^2}{x^2+y^2},$$
then what is $\displaystyle{\lim_{(x,y) \to (0,0)} f(x,y)}$?
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Solution: This limit does not exist. You can find points arbitrarily close to the origin with $f(x,y)=1$, $f(x,y)=-1$ or anything in between. For instance, $f(0.00001,0)=1$, $f(0,0.00001)=-1$, and $f(0.00001, 0.00001)=0$. (When written in polar coordinates, $f(x,y)=\cos^2(\theta)-\sin^2(\theta) = \cos(2\theta)$.) |
Just as with one variable, we say a function is continuous if it equals its limit:
| A function $f(x,y)$ is continuous at the point $(a,b)$ if $\displaystyle{\lim_{(x,y) \to(a,b)}f(x,y) = f(a,b)}$. A function is continuous on a domain $D$ if is is continuous at every point of $D$. |
Most of the rules for continuous functions carry over unchanged from single variable calculus. For instance,
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