Exploring Different Statements to Conclude Existence of f(3) | An Analysis of Math Functions

10. Which of the following statements, if true, can be used to conclude that f(3) exists?I. lim (x->3) x existsII. f is continuous at x=3III. f is differentiable at x=3A. I onlyB. II onlyC. II and III onlyD. I, II, and III

To determine which statement(s) can be used to conclude that f(3) exists, let’s break down the three statements given

To determine which statement(s) can be used to conclude that f(3) exists, let’s break down the three statements given.

I. lim (x->3) x exists
This statement implies that the limit of x as it approaches 3 exists. In other words, the function is approaching a specific value as x gets closer and closer to 3. However, this alone does not guarantee the existence of f(3). It only tells us about the behavior of the function as x approaches 3, not what happens exactly at x=3. For example, the function value itself could be undefined or have a different value at x=3.

II. f is continuous at x=3
This statement implies that the function is continuous at x=3. In order for a function to be continuous at a certain point, three conditions must be met: the function must be defined at that point, the limit of the function as x approaches that point must exist, and the limit must be equal to the value of the function at that point. Therefore, if f is continuous at x=3, it means that f(3) exists.

III. f is differentiable at x=3
This statement implies that the function is differentiable at x=3. In order for a function to be differentiable at a certain point, it must be both continuous at that point and have a defined derivative at that point. Similar to statement II, if the function is differentiable at x=3, it means that f(3) exists.

From the analysis above, we can conclude that statements II and III (f is continuous at x=3 and f is differentiable at x=3) can both be used to conclude that f(3) exists.

Therefore, the answer is C. II and III only.

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