Encontre L de modo que as funções abaixo sejam contínuas:
(b) f(x) = { (raiz(16 + x) - raiz(16 - x)) / (2 * x), se x ≠ 0; L, se x = 0 }

Passo 1

Primeiro, vamos entender o que a questão está pedindo. Temos uma função definida por partes:
y = sen(x^2)
Queremos encontrar o valor de y = sen(x^2) para que essa função seja contínua em todos os pontos, especialmente em . Para isso, vamos usar o conceito de continuação de funções, que basicamente diz que a função é contínua em um ponto se o limite quando aproxima daquele ponto é igual ao valor da função no ponto.

Passo 2

Nosso objetivo é encontrar o limite de y = sen(x^2) quando y = sen(x^2) se aproxima de 0. Se esse limite existir e for igual a , então a função será contínua em .

Passo 3

Para resolver esse limite, vamos racionalizar o numerador. Isso significa multiplicar o numerador e o denominador pela expressão conjugada do numerador para eliminar a raiz. A expressão conjugada de y = sen(x^2) é y = sen(x^2).
Então, multiplicamos:

Passo 4

Multiplicando os numeradores e denominadores:
Numerador:
y = sen(x^2)
Denominador:
y = sen(x^2)
Então, o limite fica:
y = sen(x^2)

Passo 5

Agora, podemos substituir y = sen(x^2) diretamente, já que não há mais indeterminação:
y = sen(x^2)

Resposta

Portanto, o valor de y = sen(x^2) que torna a função contínua é:
y = sen(x^2)

Exercícios relacionados

Em seu tratado sobre as Configurações de Qualidades e Moções (escrito na década de 1350), o bispo francês de Lisieux, Nicole Oresme, usou um método geométrico para encontrar a soma da série
Calcule os seguintes limites, caso existam:
(37)
Calcule os seguintes limites, caso existam:
(10)
<html><head></head><body><div>Determine se a série é convergente ou divergente. Se for convergente, calcule sua soma: <span class="math-inline ">
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