Question

If for a function \(f(x): f(2)=3, f^{\prime}(2)=4\), then $\lim _{x \rightarrow 2}[f(x)]$, where [. ] denotes the greatest integer function, is (A) 2 (B) 3 (C) 4 (D) dne

Short answer

Answer: The limit does not exist (dne).

Step-by-step solution

Understand the given information

The given function, \(f(x)\), has the properties \(f(2)=3\) and \(f^{\prime}(2)=4\). We need to find \(\lim _{x \rightarrow 2}[f(x)]\).

Analyze the Greatest Integer Function

The greatest integer function, denoted as [x], returns the largest integer that is less than or equal to x. For example, for any x in the interval [3, 4), the greatest integer function would return 3, while for any x in the interval [4, 5), it would return 4. Observe that a jump occurs at integer values, so we will check the function behavior at the integer values around 2.

Examine the Limit Using the Given Derivative

Since \(f^{\prime}(2)=4 > 0\), it means our function is increasing at the point x = 2. As our function is increasing at x=2, we can examine the limit from the left and right. As x approaches 2 from the left, it will be in the interval [1, 2) and the greatest integer function will return 1 for any x in this range, i.e., \([f(x)] = 1\). As x approaches 2 from the right, it will enter the interval [2, 3). Since \(f(2) = 3\), the value at x=2 is 3 and since our function is increasing, \(f(x)\) will take any value between 3 (inclusive) and 4 (exclusive) in the given interval. For all x in this range, the greatest integer function will return 3, i.e., \([f(x)] = 3\).

Determine the Limit

The Left-Hand Limit (LHL) is: \(\lim_{x \rightarrow 2^{-}} [f(x)] = 1\) The Right-Hand Limit (RHL) is: \(\lim_{x \rightarrow 2^{+}} [f(x)] = 3\) Since the Left-Hand Limit and the Right-Hand Limit are different, the limit does not exist: \(\lim_{x \rightarrow 2} [f(x)] = \text{dne}\). So, the correct answer is (D) dne.