Use a Definição (2.4) para provar que $\lim _{x \rightarrow 6}(5 x-21)=9$

Passo 1

Então, a gente tem que provar que o limite de quando se aproxima de 6 é igual a 9. Vamos usar a Definição 2.4, que na real é a famosa definição - de limite. Basicamente, precisamos mostrar que para qualquer número pequeno , a gente consegue encontrar um tal que se está dentro de de 6, então tá dentro de de 9. Para isso, vamos precisar manipular as inequações e encontrar uma relação entre e .

Passo 2

Primeiro, vamos escrever o que a gente quer provar usando a definição. Queremos que:
Simplificando dentro do valor absoluto:
Agora, vamos fatorar o 5:
Isso nos dá:
Então, se a gente escolher:

Passo 3

Agora, vamos verificar se essa escolha de funciona. A definição diz que para todo , existe um tal que se , então .
Substituindo por :
Se , então:
Isso mostra que nossa escolha de atende à condição necessária.

Resposta

Portanto, usando a Definição 2.4, mostramos que .

Exercícios relacionados

Esboce o gráfico da função definida por partes e, para o valor indicado, determine cada limite, se existir.
(a) (b) ©
Esboce o gráfico da função definida por partes e, para o valor indicado, determine cada limite, se existir.
(a) (b) ©
Seja definida como segue
éé
Mostre que não existe para nenhum número real .
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