Se $c>0, \lim _{x \rightarrow c} f(x)=\frac{1}{c^{2}}=f(c)$

Passo 1

Beleza! O enunciado diz que, para , o limite de quando se aproxima de é igual a , que também é igual a . Vamos usar o conceito de continuidade de funções para entender isso. Para resolver, precisamos lembrar que uma função é contínua em um ponto se o limite quando se aproxima desse ponto for igual ao valor da função nesse ponto.

Passo 2

Primeiro, vamos relembrar o que significa o limite de uma função em um ponto. Dizemos que se, conforme se aproxima de , os valores de se aproximam de .

Passo 3

Agora, para a função ser contínua em , precisamos que três condições sejam atendidas:
  1. está definida.
  2. existe.
  3. .
No enunciado, já nos dizem que e que .

Passo 4

Vamos verificar as condições:
  1. está definida? Sim, porque e , então não há divisão por zero.
  2. O limite existe? Sim, foi dado que .
  3. O limite é igual a ? Sim, ambos são .

Passo 5

Com todas as três condições satisfeitas, podemos concluir que a função é contínua em .

Resposta

A função é contínua para todo , já que .

Exercícios relacionados

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