Limits of Functions of Several Variables
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Definition of limit
Given a function \(f\) of \(n\) variables (\(f:\mathbb{R}^n \to \mathbb{R}\)), and a point \(A(a_1, a_2, \dots, a_n)\), we say that the limit of \(f(x_1, x_2, \dots, x_n)\) at \((a_1, a_2, \dots, a_n)\) is a real number \(L\), or symbolically
\[\lim_{(x_1,x_2,\dots,x_n)\to(a_1,a_2,\dots,a_n)} f(x_1, x_2, \dots, x_n) = L\]
if for every \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \(\left\lvert f(x_1,x_2,\dots,x_n) - L \right\rvert< \varepsilon\) whenever \(0 < \left\lvert\left\langle x_1, x_2, \dots, x_n\right\rangle - \left\langle a_1, a_2, \dots, a_n\right\rangle \right\rvert< \delta\).
Examples
Some examples of functions that don’t have a limit at some point.
Example 1
The surface graph of the function \(\displaystyle f(x,y) = \frac{xy}{x^2+y^2}\) from the previous video is
Example 2
Try it yourself
Some limits that exist
If a limit exists, we have number of ways to find it and prove that the number we found is really the limit.
Squeeze Theorem
The surface graph of the function \(\displaystyle f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}\) from the previous video is
The graph shows the function “squeezed” between \(\left\lvert y\right\rvert\) and \(-\left\lvert y\right\rvert\).
Replacement Theorem
The surface graph of the function \(\displaystyle f(x,y) = \frac{x^3 - y^3}{x^2 + xy + y^2}\) from the previous video is
You can see that it is really just a linear function, except that it is undefined at \((0,0)\) (you cannot see that from the graph, of course).