$\operatorname{Se} 4

Passo 1

Ok, vamos lá! O enunciado fala sobre o comportamento de uma função em torno de certos pontos. A gente vai trabalhar com limites, que fazem parte do estudo de cálculo, especificamente no tópico de continuidade de funções. Para resolver isso, vamos precisar entender o que são limites e como eles se relacionam com os valores da função nos pontos especificados.

Passo 2

Primeiro, observe que a função está definida de maneira diferente em diferentes intervalos de . Para valores de entre 4 e 8 (ou seja, ), o limite de quando se aproxima de é igual a e isso também é igual a . Isso significa que, nesse intervalo, a função é contínua porque o limite existe e coincide com o valor da função.

Passo 3

Agora, vamos olhar para os pontos extremos: quando e . No ponto , o limite de quando se aproxima de 4 é 0, e isso é igual a . Da mesma forma, no ponto , o limite é 2, que também é igual a . Isso indica que a função também é contínua nesses pontos específicos.

Passo 4

Para garantir que a função seja contínua em todo o intervalo [4, 8], precisamos definir de forma que ela se comporte conforme os limites descritos. Assim, podemos definir da seguinte maneira:
Com essa definição, garantimos que em todo o intervalo [4, 8], a função é contínua porque os limites nos pontos internos e nos extremos coincidem com os valores da função.

Resposta

A função definida por
é contínua no intervalo porque os limites em todos os pontos coincidem com os valores da função nesses pontos.

Exercícios relacionados

1 Salto 3 Removível 5 Salto
7 Infinita 9 Removível 11 Salto
13 Removível 15 Removível 17 Removível
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