$29 \lim _{x \rightarrow 0} f(x)=1 \neq 0=f(0)$
Passo 1
Beleza, vamos dar uma olhada nesse exercício. A gente tem uma função
onde o limite quando
se aproxima de 0 é 1, ou seja,
. Mas, o valor da função em 0 é diferente desse limite:
. O assunto aqui é continuidade de funções, e vamos precisar entender o que é para resolver isso.
Passo 2
Primeiro, vamos lembrar o que significa uma função ser contínua em um ponto. Uma função
é contínua em
se três condições forem atendidas:
-
está definida. - O limite de
quando se aproxima de existe. - O limite de
quando se aproxima de é igual a .
Passo 3
No nosso caso, queremos saber se
é contínua em
. Vamos verificar as três condições:
-
está definida? Sim, porque foi dado que . - O limite de
quando se aproxima de 0 existe? Sim, foi dado que . - O limite é igual a
? Aqui é que pode ter um problema. O limite é 1 e é 0.
Passo 4
Como o terceiro ponto não é atendido (porque 1 não é igual a 0), podemos concluir que a função
não é contínua em
.
Resposta
A função
não é contínua em
porque o limite de
quando
se aproxima de 0 não é igual ao valor da função em 0.
Exercícios relacionados
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Se
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