If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?I. g(x) = 1 at least onceII. lim𝑥→3.5𝑔(𝑥)=𝑔(3.5)III. lim𝑥→3−𝑔(𝑥)=lim𝑥→3+𝑔(𝑥) I. g(x) = 1 at least onceII. x→3.5lim g(x)=g(3.5)III. x→3−lim g(x)= x→3+lim g(x)
Question
If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?I. g(x) = 1 at least onceII. lim𝑥→3.5𝑔(𝑥)=𝑔(3.5)III. lim𝑥→3−𝑔(𝑥)=lim𝑥→3+𝑔(𝑥) I. g(x) = 1 at least onceII. x→3.5lim g(x)=g(3.5)III. x→3−lim g(x)= x→3+lim g(x)
Solution
I. This statement is necessarily true due to the Intermediate Value Theorem. Since g(x) is continuous for all real numbers and g(3) = -1 and g(4) = 2, there must be at least one point between x = 3 and x = 4 where g(x) = 1.
II. This statement is also necessarily true due to the definition of continuity. If a function is continuous at a point, then the limit as x approaches that point is equal to the value of the function at that point. Therefore, lim x→3.5 g(x) = g(3.5).
III. This statement is necessarily true due to the definition of continuity as well. If a function is continuous at a point, then the left-hand limit (as x approaches from the left) and the right-hand limit (as x approaches from the right) are equal. Therefore, lim x→3− g(x) = lim x→3+ g(x).
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