Cap\303\255tulo 3: Limites de V\303\241rias Vari\303\241veis
Esta planilha destina-se a examinar o material sobre os limites para fun\303\247\303\265es de 2 vari\303\241veis.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=
1-Fun\303\247\303\265es de uma vari\303\241vel, uma revis\303\243o:
Antes de examinar as defini\303\247\303\265es de limite e continuidade para fun\303\247\303\265es de v\303\241rias vari\303\241veis, vale a pena revis\303\241-las para obter as fun\303\247\303\265es de uma vari\303\241vel.
Quando dizemos que o limite quando x se aproxima de a de f (x) \303\251 L, queremos dizer que para cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= > 0 existe um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== > 0 tal que 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 sempre que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSShtZmVuY2VkR0YkNiYtRiM2Jy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRjM2LVEoJm1pbnVzO0YnRi9GNkY5RjtGPUY/RkFGQy9GRlEsMC4yMjIyMjIyZW1GJy9GSUZmbi1GUDYlUSJhRidGU0ZWLyUrZXhlY3V0YWJsZUdGOEYvRi8vJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGX29GMi1GUDYlUScmIzk0ODtGJy9GVEY4Ri9GW29GLw== .
Isso \303\251 muito bom. Para coloc\303\241-lo em termos mais visuais, estamos reivindicando que o ponto (a, L) pertence \303\240 continua\303\247\303\243o natural do gr\303\241fico de y = f (x). Testamos a afirma\303\247\303\243o colocando uma caixa em torno do ponto (a, L), subindo e descendo por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= e direita e esquerda pelo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== . Afirmamos que, independentemente do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= escolhido, podemos escolher o delta para que o gr\303\241fico saia pelas laterais da caixa de visualiza\303\247\303\243o e n\303\243o pelas partes superior e inferior.
Exemplo 1, demonstrando um limite:
Afirmamos que quando x se aproxima de 2, o limite de 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 \303\251 10. Para provar essa afirma\303\247\303\243o, precisamos de uma regra para encontrar um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== para cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= maior que 0. Estaremos satisfeitos em encontrar um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= \303\251 0,01. Come\303\247amos tentando o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== em 0,01 tamb\303\251m.
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Fica claro no gr\303\241fico que nosso LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= \303\251 muito grande. Em seguida, tentamos um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== de 0,001.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Esse valor do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== funciona. Como aumentamos o zoom o suficiente para fazer o gr\303\241fico parecer uma reta, suspeitamos que definir LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRKCYjMTAxMztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYkUSQxMDBGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQvJStleGVjdXRhYmxlR0Y0RjU= funcione para valores menores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= . Isso \303\251 um problema para outro dia, no entanto.
>
Exemplo 2, refutando um limite:
O problema inverso \303\251 mostrar que algo n\303\243o \303\251 um limite. Para fazer isso, precisamos encontrar um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= em que nenhum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== seja pequeno o suficiente. No problema acima, suponha que algu\303\251m alegou que o limite \303\251 11 e n\303\243o 10. Queremos encontrar um intervalo y em torno de 11 que o gr\303\241fico sempre se escave, n\303\243o importa qu\303\243o estreito. Veremos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= = 0,5 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== = 2.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Exemplo 3, Mostrando uma fun\303\247\303\243o n\303\243o tem limite:
Um problema mais dif\303\255cil \303\251 mostrar que n\303\243o h\303\241 limite em um ponto. Isso geralmente significa que o gr\303\241fico est\303\241 indo para o infinito ou est\303\241 tentando chegar perto de 2 pontos diferentes. Nesse caso, tornamos o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= igual a 1/3 a dist\303\242ncia entre os dois valores de y e nenhuma caixa inclui os dois.
Considere a fun\303\247\303\243o 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 . Observando o gr\303\241fico, vemos que, quando x se aproxima de 0, precisamos incluir 1 e -1. Se o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= for 2/3, n\303\243o haver\303\241 L, podemos escolher colocar 1 e -1 no intervalo [L-2/3, L + 2/3].
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZVLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YrLUYsNi1RKiZjb2xvbmVxO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZTRistRkc2JVEieEYnRkpGTUYrLUYsNi1RKCZzcmFycjtGJ0YvRjJGNUY3RjlGO0Y9Rj9GUkZURistRkc2JVEkYWJzRicvRktGNEYvLUkobWZlbmNlZEdGJDYkLUYjNiVGVS8lK2V4ZWN1dGFibGVHRjRGL0YvLUYsNi1RIi9GJ0YvRjJGNS9GOEZMRjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGZW9GVS1GLDYtUSI7RidGL0YyL0Y2RkxGN0Y5RjtGPUY/RkFGVC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZDLyUmZGVwdGhHRmBwLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZccDYmRl5wRmFwRmNwL0ZmcFElYXV0b0YnLUZHNiVRImFGJ0ZKRk1GK0ZPRistSSNtbkdGJDYkUSIwRidGL0Znb0YrLUZHNiVRIkxGJ0ZKRk1GT0YrRl9xLUYsNi1RIjpGJ0YvRjJGNUY3RjlGO0Y9Rj9GUkZURltwRmhwLUZHNiVRJGVwc0YnRkpGTUYrRk9GKy1GYHE2JFEkMS4yRidGL0ZmcUZbcEZocC1GRzYlUSRkZWxGJ0ZKRk1GK0ZPRistRmBxNiRRIy4xRidGL0ZmcUZbcEZocC1GRzYlUSVwbG90RidGSkZNLUZqbjYkLUYjNkZGRkZpbi1GLDYtUSIsRidGL0YyRmpvRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnRitGVS1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/RlJGVEZccS1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGaHNGX3ItRiw2LVEjLi5GJ0YvRjJGNUY3RjlGO0Y9Rj9GZ3NGREZccS1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/RmdzRmlzRl9yRlxzRistRkc2JVEieUYnRkpGTUZhc0ZjcUZkc0ZpcUZqc0ZjcUZddEZpcUZcc0YrLUZHNiVRJWF4ZXNGJ0ZKRk1GYXMtRkc2JVEmYm94ZWRGJ0ZKRk1GXHNGKy1GRzYlUShkaXNjb250RidGSkZNRmFzLUZHNiVGTEZKRk1GXm9GL0YvRmdvRl5vRi8= 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2-Limites de fun\303\247\303\265es em 2 vari\303\241veis:
A f\303\241cil generaliza\303\247\303\243o da defini\303\247\303\243o
Para fun\303\247\303\265es em duas vari\303\241veis, podemos prosseguir fazendo pequenas modifica\303\247\303\265es na defini\303\247\303\243o anterior. Se torna:
Quando dizemos que o limite quando (x, y) se aproxima de (a, b) de f (x, y) \303\251 L, queremos dizer que ...
para cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSIwRidGMi8lK2V4ZWN1dGFibGVHRjFGMg== existe um LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyLyUrZXhlY3V0YWJsZUdGMUYy tal que abs 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 sempre que (x, y) <> (a, b), |xa| < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== e |yb| LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHRjRGLy8lK2V4ZWN1dGFibGVHRjRGLw== .
Isso \303\251 ainda mais emocionante. Para coloc\303\241-lo em termos mais visuais, estamos reivindicando que o ponto (a, b, L) pertence \303\240 continua\303\247\303\243o natural do gr\303\241fico de y = f (x, y). Testamos a reivindica\303\247\303\243o colocando uma caixa em torno do ponto (a, b, L), subindo e descendo por epsilon e direita, esquerda, frente e volta pelo delta. Afirmamos que, independentemente do epsilon escolhido, podemos escolher o delta para que o gr\303\241fico saia pelas laterais da caixa de visualiza\303\247\303\243o e n\303\243o pelas partes superior e inferior.
(Nota avan\303\247ada: a defini\303\247\303\243o \303\251 um pouco diferente da defini\303\247\303\243o na maioria dos livros did\303\241ticos. Estamos usando bairros quadrados em vez de bairros redondos porque os bairros quadrados s\303\243o mais f\303\241ceis de desenhar.)
Exemplo 1, um limite f\303\241cil:
O caso normal com o qual lidaremos \303\251 uma fun\303\247\303\243o que possui um furo que precisamos preencher. Queremos olhar para o gr\303\241fico e ver que temos um limite \303\263bvio.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
O gr\303\241fico parece ser plano, levando-nos a supor que escolher delta igual a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= funcionar\303\241 na defini\303\247\303\243o. Voc\303\252 pode verificar se isso funcionar\303\241 se o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= for .1, .01 ou .001. De fato, essa fun\303\247\303\243o \303\251 t\303\243o bem comportada que voc\303\252 deve tentar um delta de 1 com um epsilon de 0,2 para ver que nem todo delta funciona.
Como no caso de vari\303\241vel \303\272nica, isso n\303\243o prova que a fun\303\247\303\243o tem 1 como limite quando a entrada se aproxima da origem, mas nos deixa bastante convencidos do fato.
Exemplo 2, um n\303\243o limitador f\303\241cil:
Agora nos voltamos para uma fun\303\247\303\243o que n\303\243o tem limite na origem.
Observe que observamos um patch maior ao tentar entender uma fun\303\247\303\243o mais confusa.
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A imagem nos faz suspeitar que valores z que variam de -.5 a 0.5 ocorrem arbitrariamente perto da origem. Isso significa que, quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= tiver 0,4, nossa caixa ter\303\241 apenas 0,8 de altura, mas precisar\303\241 cobrir um intervalo y que seja 1 de altura. Nenhum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== ser\303\241 pequeno o suficiente.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Voc\303\252 deve verificar se a imagem n\303\243o melhora. Fazemos o delta .01 ou 001. Parece que n\303\243o h\303\241 "maneira correta de preencher esta fun\303\247\303\243o na origem. A fun\303\247\303\243o n\303\243o tem um limite na origem.
Exemplo 3, um n\303\243o restrito r\303\255gido:
Caminhos de teste
Em nosso \303\272ltimo exemplo, tivemos linhas entrando em nosso ponto de problema (a origem) que t\303\252m limites diferentes. \303\200s vezes, quando olhamos para o gr\303\241fico, vemos que os contornos que entram na origem n\303\243o s\303\243o linhas. Considere a seguinte fun\303\247\303\243o.
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Os contornos ficam confusos perto da origem. (Os contornos parecem que est\303\243o tentando ser par\303\241bolas at\303\251 chegarem muito perto do eixo x.) Isso significa que precisamos ter cuidado com o simples zoom. Nosso grapher pode perder recursos interessantes. Considere o seguinte zoom.
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Quando aumentamos o zoom, todos os contornos desaparecem.
Quando vemos coisas estranhas acontecendo em um gr\303\241fico, devemos suspeitar que simplesmente aumentar o zoom pode perder os recursos interessantes. Uma das coisas que podemos fazer \303\251 observar o que acontece ao longo de um caminho especificado. (Hora de lembrar as curvas param\303\251tricas do pr\303\251-c\303\241lculo.) Os contornos do gr\303\241fico original parecem que est\303\243o tentando ser par\303\241bolas. Assim, veremos os gr\303\241ficos das par\303\241bolas.
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Isso deixa claro que os contornos s\303\243o par\303\241bolas. Como a origem precisaria estar em todas essas curvas de n\303\255vel, n\303\243o h\303\241 valor certo para fornecer na origem e a fun\303\247\303\243o n\303\243o tem limite l\303\241.
Isso deixa claro que as curvas est\303\243o direcionadas para pontos diferentes. Esta fun\303\247\303\243o n\303\243o tem limite na origem.
Exerc\303\255cios
1) D\303\252 evid\303\252ncia de que o limite de 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 como x se aproxima de 2 \303\251 12, encontrando valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== que funcionam na defini\303\247\303\243o quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= \303\251 0,1, 0,01. e 0,001.
2) Explique por que o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJ0YyLUYjNiUtRiw2JVEieEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRkZGSC8lLmxpbmV0aGlja25lc3NHRkAvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTi8lKWJldmVsbGVkR0YxLyUrZXhlY3V0YWJsZUdGMUYyRjJGU0Yy n\303\243o tem limite quando x se aproxima de 0. (Dica, voc\303\252 deseja explicar por que uma caixa ao redor do limite precisa incluir os valores y 1 e -1.)
3) Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQS1GPjYtUSI9RidGQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRlxvLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRI3h5RidGL0YyLUY2NiQtRiM2Jy1JJW1zdXBHRiQ2JUY6LUYjNiUtSSNtbkdGJDYkUSIyRidGQUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY+Ni1RKCZtaW51cztGJ0ZBRkNGam5GSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmpwLUZbcDYlRlhGXXBGY3BGZW5GQUZBRmVuRkEtRiM2J0Zqby1GPjYtUSIrRidGQUZDRmpuRkhGSkZMRk5GUEZpcEZbcUZccUYvRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhxLyUpYmV2ZWxsZWRHRkUtRiw2I1EhRidGZW5GQQ== quando (x, y) (0,0). Justifique a afirma\303\247\303\243o de que o limite de f (x, y) como (x, y) se aproxima de (0,0) \303\251 0, encontrando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== que funcionam na defini\303\247\303\243o do limite quando o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= \303\251 0,1, 0,01 e 0,001 .
4) Seja 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 . Justifique a afirma\303\247\303\243o de que o limite de f (x, y) como (x, y) se aproxima de (0,0) \303\251 1, encontrando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== que funcionam na defini\303\247\303\243o do limite quando o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= \303\251 0,1, 0,01 e 0,001 .
5) Seja 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 quando (x, y) se aproxima de (0,0). Mostre que a fun\303\247\303\243o n\303\243o tem um limite na origem.
6) Seja 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 quando (x, y) se aproxima de (0,0). Mostre que a fun\303\247\303\243o n\303\243o tem um limite na origem.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=