Limit and continuity of a multivariable function: Difference between revisions

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O conceito básico continua o mesmo. Porém, em 2D as coisas mais complicadas do que em 1D. Em 1D você só pode andar para frente ou para trás. Em 2D você pode andar em círculos ao redor de um ponto. O que significa que às vezes o limite pode não existir em uma direção, mas existir na outra. De acordo com o ciclo trigonométrico é razoável imaginar que existam pelo menos 360 caminhos diferentes para se chegar num ponto em 2D. Em 3D pense em ilusões ópticas. Um objeto pode parecer contínuo a partir de um certo ângulo, mas ser descontínuo em outro. Um exemplo disto é o ''Triângulo de Penrose''.
The basic concept remains the same. However, with 2D things are more complicated than in 1D. In 1D you either walk forwards or backwards. In 2D we can circle around a point, meaning that, sometimes, the limit may not exist in one direction while it does in another. According to the unit circle we'd have at least 360 different paths to take to reach the point.  


Para uma variável damos um passo para a esquerda ou para a direita. Para duas variáveis podemos dar um passo para cima ou para baixo, além da esquerda e direita. Se podemos dar um pequeno passo em qualquer direção num plano, efetivamente descrevemos um círculo. Em 3D temos uma esfera em pontos.
For one variable we take one step to the right or one step to the left. For two variables we can take one step up or down, in addition to left and right. For three variables we can take one step to the front or back, in addition to the previous directions. In 2D we have a circle of points around the point which we are evaluating a limit at. In 3D we have a sphere of points.


The idea of 360° for functions of two variables is more or less the "Penrose's Triangle"'s idea. From a certain angle it appears to be a continuous shape. However, from another angle it's revealed that the shape is discontinuous. Use a software to plot a discontinuous function of two variables and spin the viewpoint in an attempt to "hide" the discontinuity.


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Em 2D temos uma equação de um círculo e em 3D uma esfera. A equação da circunferência é: <math>(x - x_0)^2 + (y - y_0)^2 = \delta^2 \iff \delta = \sqrt{(x - x_0)^2 + (y - y_0)^2}</math> ''(não estamos interessados num raio negativo. Podemos ignorar a raiz negativa)''. A equação também é a distância entre a origem da circunferência e <math>P_2</math>. Suponha que <math>P = (a,b)</math> esta localizado em qualquer ponto do círculo, exceto pelo perímetro. A sua imagem, <math>f(P)</math> estará localizada em qualquer lugar no intervalo <math>\left[L - \epsilon, L + \epsilon\right]</math>.
We have an equation of a circle in 2D and the equation of a sphere in 3D. The equation for the circle is: <math>(x - x_0)^2 + (y - y_0)^2 = \delta^2 \iff \delta = \sqrt{(x - x_0)^2 + (y - y_0)^2}</math> ''(we aren't interested in a negative radius, we can disregard the negative root)''. That equation is also the distance between the circle's origin and <math>P_2</math>. Suppose that <math>P = (a,b)</math> is located anywhere in that circle, excluding the circle's perimeter. Its image, <math>f(P)</math> is going to be located anywhere in <math>\left[L - \epsilon, L + \epsilon\right]</math>.


Note também como a figura representa graficamente esta propriedade: <math>|a - b| = \sqrt{(a - b)^2}</math>. A distância não pode ser negativa. Podemos ver as coordenadas dos pontos como vetores de deslocamento. Tanto <math>P_2</math> quanto a origem do círculo são pontos que são deslocados da origem do plano cartesiano para as suas respectivas posições mostradas no gráfico ''(soma de vetor com ponto)''. Vetorialmente podemos interpretar o radio do círculo assim <math>||\overrightarrow{P_2} - \overrightarrow{C}|| = \delta</math>, onde <math>\overrightarrow{C}</math> é o vetor deslocamento da origem do círculo.
Notice how the figure is also a graphical depiction of the property: <math>|a - b| = \sqrt{(a - b)^2}</math>. Distance cannot be negative. We can view the coordinates of the points as displacement vectors. Both points, <math>P_2</math> and the circle's origin, being displaced from the origin of the Cartesian plane to their positions shown in the graph ''(sum a vector with a point)''. The radius of that circle can also be interpreted as <math>||\overrightarrow{P_2} - \overrightarrow{C}|| = \delta</math>, where <math>\overrightarrow{C}</math> is displacement vector for the circle's origin.


'''A notação é a mesma usada com limites de funções de uma variável:'''
'''Notation is really the same idea from limits of single variable functions:'''


<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L</math> ''(para mais de duas variáveis é a mesma coisa)''
<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L</math> ''(same for any number of variables)''


Para cada <math>\epsilon > 0</math> há um um <math>\delta > 0</math>, tal que cada <math>(x,\ y) \in D_f</math>, <math>0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta \implies |f(x,y) - L| < \epsilon</math>. ''(alguns livros substituem a raiz quadrada pela norma e a diferença entre as coordenadas. É a mesma coisa)''
For each <math>\epsilon > 0</math>, there is a <math>\delta > 0</math>, such that every <math>(x,\ y) \in D_f</math>, <math>0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta \implies |f(x,y) - L| < \epsilon</math>. ''(some textbooks replace the square root with a norm and difference between the coordinates, it's really the same thing)''


O conceito é virtualmente o mesmo do caso das funções de uma variável. Estamos considerando a menor distância entre dois pontos em 2D que seja o mais próximo possível do zero. Enquanto o erro, a distância entre a imagem e o limite, é o menor valor possível. Note que a definição do limite para várias variáveis não considera o caminho até o ponto. O conceito de limites laterais para várias variáveis é um pouco mais complicado porque em 2D e 3D temos para cima e para baixo, frente e trás e há muito mais lados e direções para serem analisados.
The concept is virtually the same used for single variable functions. We are considering the smallest distance between two points in 2D that is as close as possible to zero. While the error, the distance between the image and the limit, is the lowest possible value. Note that the definition of a limit for many variables is not considering the path to the point. The concept of one sided limits for many variables is a bit more complicated because when we expand to 2D and 3D there is up and down, front and back, there are way more sides and directions to account for.


'''Terminologia:''' quando um limite existe num ponto <math>(x_0, y_0)</math>, aquele ponto é chamado de ponto limite ou ponto de acumulação. Este termo vem da topologia.
'''Terminology:''' when a limit does exist at a point <math>(x_0, y_0)</math>, that point is called a limit point or an accumulation point. That term comes from topology.


'''Continuidade:''' a discussão é exatamente a mesma para uma ou várias variáveis.  
'''Continuity:''' the discussion is exactly the same for one or many variables.  
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<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,\ y) = f(x_0,\ y_0)</math>
<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,\ y) = f(x_0,\ y_0)</math>
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Se a função é definida em <math>(x_0, y_0)</math> e o limite converge para aquele ponto, a função é contínua ali. Estenda o mesmo raciocínio para qualquer ponto <math>(x, y)</math> escolhido do domínio da função e a função é contínua em todo o seu domínio.
If the function is defined at <math>(x_0, y_0)</math> and the limit converges to that point, the function is continuous at that point. Extend the same reasoning to any arbitrary <math>(x, y)</math> point chosen from the function's domain and the function is continuous everywhere in its domain.


==Teorema do confronto para várias variáveis==
==Squeeze theorem for many variables==


É o mesmo conceito para variáveis de uma variável. Não há diferenças.
It's the same concept from single variable functions, there is no difference.


Se <math>f(x,\ y) \leq g(x,\ y) \leq h(x,\ y)</math> for <math>0 \leq \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta</math> e
If <math>f(x,\ y) \leq g(x,\ y) \leq h(x,\ y)</math> for <math>0 \leq \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta</math> and


<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L = \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} h(x,\ y)</math>
<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L = \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} h(x,\ y)</math>


Então
Then


<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} g(x,\ y) = L</math>
<math class="big">\lim_{(x,\ y) \ \to \ (x_0,\ y_0)} g(x,\ y) = L</math>


A mesma propriedade dos limites para uma variável se aplica para várias variáveis. Se uma função limitada é multiplicada por outra, seujo limite vai para zero, podemos dizer que o limite do produto também é zero.
The same property of limits for one variable applies to many variables. If a bounded function is multiplied by another with a limit that goes to zero, we can say that the limit of the product is zero too.


==Limites de "múltiplos lados"==
=="Multi-sided" limits==
 
É inviável calcular o limite cem vezes só para verificar se ele existe ou não. Precisamos de algo para tratar deste caso. Nós temos uma ferramenta e são as '''equações paramétricas''' que descrevem trajetórias. Quando traçamos curvas de nível para funções de duas variáveis temos um caminho e uma equação com duas variáveis, sob a condição especial de que a equação é igual a um nível constante. O que estamos procurando não é exatamente uma função vetorial que descreva uma trajetória no espaço, que no caso seria um caminho sobre a superfície da função de duas variáveis. Nosso problema é que temos um ponto inicial em qualquer lugar no domínio da função, o plano XY ou parte deste, e queremos andar em direção a outro ponto, o ponto que queremos saber se o limite existe ou não. É difícil de imaginar, mas cada passo que dermos em qualquer direção no domínio da função é refletido no gráfico da mesma.


It's impractical to calculate a limit a hundred times just to check whether it exists or not. We need something to deal with this scenario. It turns out that we have to resort to the knowledge of trajectories, '''parametric equations'''. When we plot level curves for functions of two variables we have a path and an equation with two variables, under the special condition that the equation always keep the level a constant. What we are looking for isn't really a vector valued function that describes a trajectory in 3D, the path over the surface of the two variable function. Our problem is that we have a starting point anywhere on the function's domain, the XY plane or part of it, and we wish to walk towards another point, the point that we wish to check whether the limit exists or not. It's hard to picture it, but every step we take in any direction on the function's domain, is reflected on the function's graph.


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[[file:limit_paths.png|300px]]
[[file:limit_paths.png|300px]]


''(Observe que temos duas retas sobre o domínio, paralelas à cada eixo, para chegar naquele ponto. Por outro lado, na superfície da função a trajetória não é plana. A menos que a superfície da função seja plana em si)''
''(Note that we have two straight lines on the domain, parallel to each axis, to reach that point. On the function's surface the trajectories aren't flat though, unless the function itself is a plane)''
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Quais são os caminhos mais fáceis de verificar? Os mais óbvios são os próprios eixos X e Y. Igualando uma variável a zero reduzimos o limite de duas variáveis para uma só. O próximo caminho é igualando uma variável à outra <math>x = y</math>, o que significar andar sobre a diagonal do plano cartesiano. Um outro caminho muito usado é <math>y = x^2</math>. Uma parábola.
Which are the easiest paths to check? The XY axis are the most obvious ones. We keep one variable equal to zero and reduce the limit of two variables to a single variable. The next one is to make one variable equal to the other <math>x = y</math>, which translates to walking over the diagonal of the XY plane. Another common and easy path to take is <math>y = x^2</math> which is a parabola.
 
Eu tenho um livro que descreve o mesmo conceito de um modo um pouco mais complicado. No lugar de considerar <math>x = y</math> por exemplo, ele considera uma '''curva''' <math>\gamma(t) = (x(t), \ y(t))</math>. É uma função de uma variável, neste caso o tempo, que na saída produz posições no plano cartesiano. É por isto que podemos pegar <math>f(x,y)</math> e escrever <math>f(\gamma(t))</math>. Não ''"apagamos"'' as duas variáveis. Apenas as ''"escondemos"'' numa outra função que traça trajetórias no plano XY. Alguns livros trazem exemplos com a notação assim <math>f(t,t^2)</math>. É fundamentalmente o mesmo conceito que escrever <math>f(x, y=x^2)</math>.


De certa forma estamos aplicando uma técnica que olha para o problema do limite de uma função de duas variáveis e impõe algumas condições que nos permitem tratar o limite como se fosse um caso de uma função de uma variável. Quando fazemos <math>x = 0</math> ou <math>y = 0</math> não estamos exatamente transformando uma função de duas variáveis em uma função de uma só. O que estamos fazendo na verdade é ''"cortando"'' a função com um plano e olhando a sua ''"silhueta"'' deixada pela intersecção da função com o plano. Esta ''silhueta'' pode ser interpretada como uma função de uma variável porque é, na realidade, uma linha com espessura nula.
I have a textbook that describes the same concept with a slightly more complicated way. Rather than considering <math>x = y</math> for example, it considers a '''curve''' <math>\gamma(t) = (x(t), \ y(t))</math>. It's a function of one variable, in this case time, that outputs positions on the Cartesian plane. That's why we can take <math>f(x,y)</math> and write <math>f(\gamma(t))</math>. We didn't ''"erase"'' the two variables, we just ''"hid"'' them in another function that traces trajectories on the XY plane. Some examples out there and in some textbooks may have this expression <math>f(t,t^2)</math>. It's really the same concept of writing <math>f(x, y=x^2)</math>.


Eu vou mostrar alguns gráficos feitos com o Geogegra 3D para ajudar a ter uma visão melhor sobre os limites de uma função de duas variáveis:
From a certain perspective we are using a technique that is looking at the problem of a limit of a function of two variables by imposing certain conditions that allow us to treat it as a limit of a single variable function. When we make <math>x = 0</math> or <math>y = 0</math> we aren't really transforming a function of two variables into a single variable one. What we are doing is ''"slicing"'' the function with a plane and looking at its ''"silhouette"'' left by intersecting it with that plane. That ''silhouette'' can be interpreted as a single variable function because it's really a line with zero thickness.


I'm going to show some graphs plotted with Geogebra 3D to help get a better view of limits of functions of two variables:


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Vamos pegar um caso clássico de função de uma variável <math>f(x) = \frac{1}{x}</math> e estender para 3D, com a segunda variável mantida constante. Observe que neste caso o comportamento da função é exatamente o mesmo do caso de uma variável. Aproximando-se pela esquerda e pela direita resulta em limites diferentes.
Let's take the classic single variable function <math>f(x) = \frac{1}{x}</math> and plot it in 3D by keeping the second variable a constant. Note that in this case, the function's behaviour is exactly the same as its single variable counterpart. Approaching from the left or right is going to yield different limits.




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Agora vamos ver o caso de <math>f(x,y) = \frac{x^2}{x^2 + y^2}</math>. O eixo azul é o Z, o vermelho é o X e o verde é o Y. Observe que se andarmos ao longo dos eixos X e Y, os limites na origem estão em coordenadas Z diferentes. Se andarmos ao longo das diagonais, mais uma altura Z diferente quando nos aproximamos da origem. É impossível para a função ter valores diferentes para um mesmo ponto do seu domínio. O gráfico é renderizado como se fosse uma superfície contínua, mas isto é porque é impossível renderizar com uma resolução infinita. A descontinuidade na origem tem o tamanho de um pixel.
Now let's take a look at <math>f(x,y) = \frac{x^2}{x^2 + y^2}</math>. The blue axis is Z, red is X and green is Y. Notice that if we walk along the X and Y axis the limits at the origin are at a different Z coordinate. If we walk along the diagonals, yet another different Z close to the origin. It's impossible for the function to have different values for the same point in its domain. The graph is rendered as a continuous surface but this is because it's impossible to render at infinite resolution. The discontinuity at the origin has the size of a pixel.


Podemos aplicar o mesmo raciocínio para três ou mais variáveis tranquilamente. Os livros que eu conheço não trazem exercícios de limites para funções de mais do que duas variáveis. Eu suponho que acaba sendo um processo cansativo quando se tem caminhos em 3D e além. Fica impraticável fazer à mão.
We can very well extend the same reasoning to three and more variables. The textbooks that I know don't have exercises of limits for more than two variables. I suppose it's just an exhausting process to consider even more directions and paths in 3D and beyond.

Latest revision as of 20:57, 15 August 2022

The basic concept remains the same. However, with 2D things are more complicated than in 1D. In 1D you either walk forwards or backwards. In 2D we can circle around a point, meaning that, sometimes, the limit may not exist in one direction while it does in another. According to the unit circle we'd have at least 360 different paths to take to reach the point.

For one variable we take one step to the right or one step to the left. For two variables we can take one step up or down, in addition to left and right. For three variables we can take one step to the front or back, in addition to the previous directions. In 2D we have a circle of points around the point which we are evaluating a limit at. In 3D we have a sphere of points.

The idea of 360° for functions of two variables is more or less the "Penrose's Triangle"'s idea. From a certain angle it appears to be a continuous shape. However, from another angle it's revealed that the shape is discontinuous. Use a software to plot a discontinuous function of two variables and spin the viewpoint in an attempt to "hide" the discontinuity.

We have an equation of a circle in 2D and the equation of a sphere in 3D. The equation for the circle is: [math]\displaystyle{ (x - x_0)^2 + (y - y_0)^2 = \delta^2 \iff \delta = \sqrt{(x - x_0)^2 + (y - y_0)^2} }[/math] (we aren't interested in a negative radius, we can disregard the negative root). That equation is also the distance between the circle's origin and [math]\displaystyle{ P_2 }[/math]. Suppose that [math]\displaystyle{ P = (a,b) }[/math] is located anywhere in that circle, excluding the circle's perimeter. Its image, [math]\displaystyle{ f(P) }[/math] is going to be located anywhere in [math]\displaystyle{ \left[L - \epsilon, L + \epsilon\right] }[/math].

Notice how the figure is also a graphical depiction of the property: [math]\displaystyle{ |a - b| = \sqrt{(a - b)^2} }[/math]. Distance cannot be negative. We can view the coordinates of the points as displacement vectors. Both points, [math]\displaystyle{ P_2 }[/math] and the circle's origin, being displaced from the origin of the Cartesian plane to their positions shown in the graph (sum a vector with a point). The radius of that circle can also be interpreted as [math]\displaystyle{ ||\overrightarrow{P_2} - \overrightarrow{C}|| = \delta }[/math], where [math]\displaystyle{ \overrightarrow{C} }[/math] is displacement vector for the circle's origin.

Notation is really the same idea from limits of single variable functions:

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L }[/math] (same for any number of variables)

For each [math]\displaystyle{ \epsilon \gt 0 }[/math], there is a [math]\displaystyle{ \delta \gt 0 }[/math], such that every [math]\displaystyle{ (x,\ y) \in D_f }[/math], [math]\displaystyle{ 0 \lt \sqrt{(x - x_0)^2 + (y - y_0)^2} \lt \delta \implies |f(x,y) - L| \lt \epsilon }[/math]. (some textbooks replace the square root with a norm and difference between the coordinates, it's really the same thing)

The concept is virtually the same used for single variable functions. We are considering the smallest distance between two points in 2D that is as close as possible to zero. While the error, the distance between the image and the limit, is the lowest possible value. Note that the definition of a limit for many variables is not considering the path to the point. The concept of one sided limits for many variables is a bit more complicated because when we expand to 2D and 3D there is up and down, front and back, there are way more sides and directions to account for.

Terminology: when a limit does exist at a point [math]\displaystyle{ (x_0, y_0) }[/math], that point is called a limit point or an accumulation point. That term comes from topology.

Continuity: the discussion is exactly the same for one or many variables.

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,\ y) = f(x_0,\ y_0) }[/math]

If the function is defined at [math]\displaystyle{ (x_0, y_0) }[/math] and the limit converges to that point, the function is continuous at that point. Extend the same reasoning to any arbitrary [math]\displaystyle{ (x, y) }[/math] point chosen from the function's domain and the function is continuous everywhere in its domain.

Squeeze theorem for many variables

It's the same concept from single variable functions, there is no difference.

If [math]\displaystyle{ f(x,\ y) \leq g(x,\ y) \leq h(x,\ y) }[/math] for [math]\displaystyle{ 0 \leq \sqrt{(x - x_0)^2 + (y - y_0)^2} \lt \delta }[/math] and

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L = \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} h(x,\ y) }[/math]

Then

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} g(x,\ y) = L }[/math]

The same property of limits for one variable applies to many variables. If a bounded function is multiplied by another with a limit that goes to zero, we can say that the limit of the product is zero too.

"Multi-sided" limits

It's impractical to calculate a limit a hundred times just to check whether it exists or not. We need something to deal with this scenario. It turns out that we have to resort to the knowledge of trajectories, parametric equations. When we plot level curves for functions of two variables we have a path and an equation with two variables, under the special condition that the equation always keep the level a constant. What we are looking for isn't really a vector valued function that describes a trajectory in 3D, the path over the surface of the two variable function. Our problem is that we have a starting point anywhere on the function's domain, the XY plane or part of it, and we wish to walk towards another point, the point that we wish to check whether the limit exists or not. It's hard to picture it, but every step we take in any direction on the function's domain, is reflected on the function's graph.

(Note that we have two straight lines on the domain, parallel to each axis, to reach that point. On the function's surface the trajectories aren't flat though, unless the function itself is a plane)

Which are the easiest paths to check? The XY axis are the most obvious ones. We keep one variable equal to zero and reduce the limit of two variables to a single variable. The next one is to make one variable equal to the other [math]\displaystyle{ x = y }[/math], which translates to walking over the diagonal of the XY plane. Another common and easy path to take is [math]\displaystyle{ y = x^2 }[/math] which is a parabola.

I have a textbook that describes the same concept with a slightly more complicated way. Rather than considering [math]\displaystyle{ x = y }[/math] for example, it considers a curve [math]\displaystyle{ \gamma(t) = (x(t), \ y(t)) }[/math]. It's a function of one variable, in this case time, that outputs positions on the Cartesian plane. That's why we can take [math]\displaystyle{ f(x,y) }[/math] and write [math]\displaystyle{ f(\gamma(t)) }[/math]. We didn't "erase" the two variables, we just "hid" them in another function that traces trajectories on the XY plane. Some examples out there and in some textbooks may have this expression [math]\displaystyle{ f(t,t^2) }[/math]. It's really the same concept of writing [math]\displaystyle{ f(x, y=x^2) }[/math].

From a certain perspective we are using a technique that is looking at the problem of a limit of a function of two variables by imposing certain conditions that allow us to treat it as a limit of a single variable function. When we make [math]\displaystyle{ x = 0 }[/math] or [math]\displaystyle{ y = 0 }[/math] we aren't really transforming a function of two variables into a single variable one. What we are doing is "slicing" the function with a plane and looking at its "silhouette" left by intersecting it with that plane. That silhouette can be interpreted as a single variable function because it's really a line with zero thickness.

I'm going to show some graphs plotted with Geogebra 3D to help get a better view of limits of functions of two variables:

Let's take the classic single variable function [math]\displaystyle{ f(x) = \frac{1}{x} }[/math] and plot it in 3D by keeping the second variable a constant. Note that in this case, the function's behaviour is exactly the same as its single variable counterpart. Approaching from the left or right is going to yield different limits.


Now let's take a look at [math]\displaystyle{ f(x,y) = \frac{x^2}{x^2 + y^2} }[/math]. The blue axis is Z, red is X and green is Y. Notice that if we walk along the X and Y axis the limits at the origin are at a different Z coordinate. If we walk along the diagonals, yet another different Z close to the origin. It's impossible for the function to have different values for the same point in its domain. The graph is rendered as a continuous surface but this is because it's impossible to render at infinite resolution. The discontinuity at the origin has the size of a pixel.

We can very well extend the same reasoning to three and more variables. The textbooks that I know don't have exercises of limits for more than two variables. I suppose it's just an exhausting process to consider even more directions and paths in 3D and beyond.

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