Explique por que a função é descontínua no ponto a dado. Esboce o gráfico da função: f(x) = { los(x) se x < 0, 0 se x = 0, 1 - x^2 se x > 0 } a = 0

Passo 1

Beleza, vamos lá! O exercício pede para a gente explicar por que a função é descontínua no ponto e também fazer um esboço do gráfico dessa função. A matéria que vamos mexer aqui é de continuidade de funções, especificamente analisar os limites laterais e o valor da função no ponto. Para resolver, vamos precisar entender como calcular limites e comparar eles com o valor da função no ponto em questão.

Passo 2

Primeiro, vamos olhar para a definição da função :
Parece que há um pequeno erro de digitação em "los ". Provavelmente, a intenção era escrever , então vamos assumir que .

Passo 3

Agora, para verificar a continuidade em , precisamos calcular o limite da função quando se aproxima de 0 pelo lado esquerdo ( ) e pelo lado direito ( ), e comparar com o valor da função em .

Passo 4

Calculando o limite pela esquerda ( ):
Quando , . Sabemos que:

Passo 5

Calculando o limite pela direita ( ):
Quando , . Então:

Passo 6

Valor da função em :
Pela definição, .

Passo 7

Analisando a continuidade:
Para a função ser contínua em , os limites laterais devem ser iguais e também iguais ao valor da função no ponto. Aqui temos:
Como os limites laterais não são iguais ( ), a função apresenta uma descontinuidade em .

Passo 8

Esboçando o gráfico:
  • Para : , que é uma onda suave oscilando entre -1 e 1.
  • Em : , marcando um ponto no centro.
  • Para : , que é uma parábola invertida, começando em 1 quando e descendo conforme aumenta.
No ponto , há uma ruptura: a função se aproxima de 0 pelo lado esquerdo e de 1 pelo lado direito, mas só passa por 0 exatamente em .

Resposta

A função é descontínua no ponto porque os limites laterais não são iguais ( e ) e diferem do valor da função em . O esboço do gráfico mostra uma onda de seno para , um ponto em e uma parábola invertida para , com uma ruptura no ponto .

Exercícios relacionados

Calcule o limite . Existem assíntotas verticais ou horizontais? Em caso afirmativo escreva a(s) equação(ões) da(s) assíntota(s).
Calcule as integrais:
(m)
Como você
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