Question
Find a so that lim _(x arrow 1)(x) exist when [ f(x)={ 3 x+5, & x leq 1 2 x+1, & x geqslant 1 . ]
Solution
4.4
(373 Votos)
Leandra
Elite · Tutor por 8 anos
Resposta
To ensure that the limit of \( f(x) \) as
approaches 1 exists, we need to make sure that the left-hand limit and the right-hand limit are equal.First, let's find the left-hand limit as
approaches 1 from the left:
Next, let's find the right-hand limit as
approaches 1 from the right:
For the limit to exist, the left-hand limit and the right-hand limit must be equal. Therefore, we need to find a value of
such that:
Since the left-hand limit is 8 and the right-hand limit is 3, we need to set
to be the value that both limits approach. In this case, there is no value of
that can make the left-hand limit and the right-hand limit equal, as they are already fixed at 8 and 3 respectively.Therefore, the limit of \( f(x) \) as
approaches 1 does not exist.