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Which of the following statements would be correct to use when proving that limx→4(3x−4)=8?
a. Given 0<∣∣x−4∣∣<ϵ, then ∣∣(3x−4)−8∣∣<ϵ3.
b. Given 0<∣∣x−4∣∣<ϵ3, then ∣∣(3x−4)−8∣∣<ϵ.
c. Given 0<∣∣x−8∣∣<ϵ, then ∣∣(3x−4)−4∣∣<ϵ3.
d. Given 0<∣∣x−8∣∣<ϵ3, then ∣∣(3x−4)−4∣∣<ϵ.
e. Given 0<∣∣x−4∣∣<3ϵ, then ∣∣(3x−4)−8∣∣<ϵ.
f. Given 0<∣∣x−4∣∣<3ϵ, then ∣∣(3x−4)−8∣∣<ϵ3.

Answer :

Answer:

Option c

Step-by-step explanation:

given that limit x tending to 4 of the function (3x-4) is 8

This implies for all values of x such that for epsilon >0 arbitrary small ,

[tex]||x-4||<\epsilon[/tex], we get

|f(x)-8|<3epsilon

this is equivalent to the option c.

Proof:

Consider

[tex]||x-4||<\epsilon\\3||x-4||<3\epsilon\\||3x-12||<3\epsilon\\||3x-4|-8| <3\epsilon[/tex]

Hence it follows that option C is right.

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