Section 3.2 Limit and Continuity
3.2 Limit and Continuity
In this section, we learn what does it mean for a function of two variables has a limit of a given point [latex](a,b)[/latex] and what does it mean that it is continuous at a given point. The reason for understand those two concept is to decide if we can take the derivative of a function or integrate it. Recall that if [latex]y=f(x)[/latex] is a function a variable, we say the limit of the function at [latex]x=a[/latex] exits if we approach the function from [latex]x=a^{+}[/latex] and from [latex]x=a^{-}[/latex], we get the same value. The idea is the same for function of two variables but we need to approach the function from any direction of a point [latex](x,y)=(a,b)[/latex].
Definition
Let [latex]f[/latex] be a function of two variables, [latex]x[/latex] and [latex]y[/latex]. The limit of [latex]f(x,y)[/latex] as [latex](x,y)[/latex] approaches [latex](a,b)[/latex] is [latex]M[/latex], written \[\lim_{(x,y)\rightarrow(a,b)}f(x,y)=M\] if for each [latex]\varepsilon\gt 0[/latex] there exists a small enough [latex]\delta\gt 0[/latex] such that for all points [latex](x,y)[/latex] in a [latex]\delta[/latex] disk around [latex](a,b)[/latex], except possibly for [latex](a,b)[/latex] itself, the value of [latex]f(x,y)[/latex] is no more than [latex]\varepsilon[/latex] away from [latex]M[/latex]. Using symbols, we write the following: For any [latex]\varepsilon\gt 0[/latex], there exists a number [latex]\delta\gt 0[/latex] such that [latex]|f(x,y)-M|\lt \varepsilon[/latex] whenever [latex]0\lt \sqrt{(x-a)^{2}+(y-b)^{2}}\lt \delta[/latex].
Note that for a function [latex]y=f(x)[/latex], to show its limit does not exist at a given [latex]x=a[/latex], we just need to show [latex]\lim_{x\rightarrow a^{+}}f(x)=M_{1}\neq M_{2}=\lim_{x\rightarrow a^{-}}f(x)[/latex]. We do similar approach for the function of two variables.
Remark: If [latex]f(x,y)[/latex] approaches [latex]M_{1}[/latex] as [latex](x,y)[/latex] approaches [latex](a,b)[/latex] along a path [latex]C_{1}[/latex] and [latex]f(x,y)[/latex] approaches [latex]M_{2}\neq M_{1}[/latex] as [latex](x,y)[/latex] approaches [latex](a,b)[/latex] along a path [latex]C_{2}\neq C_{1}[/latex] then we say the limit of [latex]f(x,y)[/latex] at [latex](x,y)=(a,b)[/latex] does not exist. Notice that [latex](a,b)[/latex] must be on [latex]C_{1}[/latex] and [latex]C_{2}[/latex].
Example 1: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{x^{3}-3y^{2}}{x^{2}+y^{2}}\]
Exercise 1: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{2x^{2}+3y^{2}}{x^{2}+y^{2}}\]
Example 2: Find the limit
\[\lim_{(x,y)\rightarrow(0,2)}\frac{xy-2x}{x^{2}+(y-2)^{2}}\]
Exercise 2: Find the limit
\[\lim_{(x,y)\rightarrow(-1,0)}\frac{xy+y}{(x+1)^{2}+y^{2}}\]
Remark: To show the limit exits for a function at a particular point is not easy. Often time, we use many properties that are already proven that we can use them to show the limit exists. The reason to show the limit exist is difficult in two variables case is that even we show the limit of the function at many paths agree to each other, it does NOT show the limit exists because there are infinitely many paths that we need to check in order to show the limit exists.
Recall in calculus I, we can find the limit of a function using squeeze theorem. Here we can use similar approach as well.
Squeeze Theorem: Suppose [latex]g(x,y)\leq f(x,y)\leq h(x,y)[/latex] on [latex]D[/latex] inside [latex]\mathbb{R}^{2}[/latex] and let [latex](x,y)=(a,b)[/latex] be a point in [latex]D[/latex] such that \[\lim_{(x,y)\rightarrow(a,b)}g(x,y)=M=\lim_{(x,y)\rightarrow(a,b)}h(x,y)\] then \[\lim_{(x,y)\rightarrow(a,b)}f(x,y)=M\]
Example 3: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{x^{2}y}{x^{2}+y^{2}}\]
Exercise 3: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{xy^{2}}{x^{2}+y^{2}}\]
Example 4: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{x^{2}y^{4}}{x^{2}+y^{4}}\]
Exercise 4: Find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{x^{4}y^{3}}{x^{4}+y^{4}}\]
Recall in calculus I, we are able to find the limit of a function [latex]f(x)[/latex] at [latex]x=a[/latex] if we know the function is continuous at [latex]x=a[/latex]. At here, we have similar approach.
Definition: A function [latex]f(x,y)[/latex] is continuous at [latex](x,y)=(a,b)[/latex] if \[\lim_{(x,y)\rightarrow(a,b)}f(x,y)=f(a,b).\] We say [latex]f(x,y)[/latex] is continuous on [latex]D[/latex] if [latex]f(x,y)[/latex] is continuous at every point [latex]D[/latex].
Remark: All polynomials are continuous on [latex]\mathbb{R}^{2}[/latex] and rational and algebraic functions are continuous on their domain. This means that we can direct plug the value of [latex](x,y)[/latex] into the function to obtain the limit as long as the function is continuous on the point.
Example 5: Find the limit
\[\lim_{(x,y)\rightarrow(-1,0)}\frac{x^{3}-3y^{2}}{x^{2}+y^{2}+1}\]
Exercise 5: Find the limit
\[\lim_{(x,y)\rightarrow(1,1)}\frac{x^{3}-3y^{2}+xy}{x^{2}-y^{2}+3y+2}\]
Remark: It can be shown that if [latex]f(x,y)[/latex] is a continuous function of two variables and [latex]g(z)[/latex] is a continuous function of a single variable that is defined on the range of [latex]f(x,y)[/latex], then the composite function [latex]h(x,y)=g(f(x,y))[/latex] is also a continuous function.
Example 6: Find the limit
\[\lim_{(x,y)\rightarrow(2,1)}\text{sin}(\sqrt{x+y-1})\]
Exercise 6: Find the limit
\[\lim_{(x,y)\rightarrow(1,3)}\text{cos}(\frac{xy}{2x+y})\]
Example 7: Find the limit
\[\lim_{(x,y,z)\rightarrow(0,0,0)}\frac{x^{2}+y^{2}-z^{2}}{xy-yz}\]
Example 8: Use polar coordinate to find the limit
\[\lim_{(x,y)\rightarrow(0,0)}\frac{x^{3}+y^{3}}{x^{2}+y^{2}}\]
Group work:
1. Find the limit \[\lim_{(x,y)\rightarrow(0,0)}\frac{2xy}{3x^{2}+y^{2}}\]
2. Find the limit \[\lim_{(x,y)\rightarrow(0,0)}\frac{xy^{4}}{x^{2}+y^{8}}\]
3. Use polar coordinate to find the limit \[\lim_{(x,y)\rightarrow(0,0)}\frac{2x^{2}+3y^{2}}{x^{2}+y^{2}}\]
4. Find the limit \[\lim_{(x,y,z)\rightarrow(0,0,0)}\frac{x^{2}y^{2}z^{2}}{x^{2}+y^{8}}\]
5. Find the limit \[\lim_{(x,y,z)\rightarrow(0,0,0)}\frac{xy+yz+xz}{x^{2}+y^{2}+z^{2}}\]