Calcule os seguintes limites, caso existam: (35) $\lim _{x \rightarrow-7} \frac{3^{(x+7) / 5}-1}{x+7}$
Passo 1
Vamos lá! A questão pede para calcular o limite:
Estamos trabalhando com cálculo de limites, especificamente para funções exponenciais. Para resolver isso, vamos precisar entender como simplificar esse tipo de expressão quando o denominador se aproxima de zero.
Passo 2
Primeiro, vamos verificar o tipo de limite que temos aqui. Quando
se aproxima de -7, o termo
se aproxima de 0. Portanto, tanto o numerador quanto o denominador estão se aproximando de 0, resultando em uma forma indeterminada do tipo
.
Quando encontramos uma forma indeterminada
, uma técnica útil é a Regra de L’Hospital, que nos permite calcular o limite derivando o numerador e o denominador separadamente.
Passo 3
Vamos aplicar a Regra de L’Hospital:
Primeiro, calculamos a derivada do numerador em relação a
:
A derivada do denominador
em relação a
é simplesmente 1.
Agora, aplicando a Regra de L’Hospital, o limite original se transforma em:
Resposta
O limite é
.
Exercícios relacionados
Calcule os seguintes limites, caso existam:
(23)
(23)
Determine se a série é convergente ou divergente. Se for convergente, calcule sua soma:
<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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Calcule os seguintes limites, caso existam:
(37)
(37)