Question

\(\lim _{\mathrm{x} \rightarrow \pi}\\{(\sin 3[\mathrm{x}]) /[\mathrm{x}]\\}=?\) \\{Where []\(=\) greatest integer part \(\mathrm{x} \in(\pi-0.01, \pi+0.01)\\}\) (a) 3 (b) \([(\sin 3) / 9]\) (c) \([(\sin 9) / 3]\) (d) \(\sin 9\)

Short answer

(c) \(\frac{\sin(9)}{3}\)

Step-by-step solution

Define the function

Let's define the function: \(f(x) = \frac{(\sin(3\lfloor x \rfloor))}{(\lfloor x \rfloor)}\) where the \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). We are interested in finding the limit of this function as \(x\) approaches \(\pi\). More specifically, we want to find the value of \(\lim _{x \rightarrow \pi} f(x)\).

Analyze the properties of \(f(x)\) near \(x = \pi\)

The greatest integer less than or equal to \(x\) will be the same for all \(x\) in the interval \((\pi - 0.01, \pi + 0.01)\), since \(\pi\) is roughly \(3.14\). So, the greatest integer part of \(x\) will be \(\lfloor x \rfloor = 3\), for all such \(x\). Thus, for all such \(x\), we have: \(f(x) = \frac{(\sin(3 \cdot 3))}{3}\) Notice that this function doesn't depend on the variable \(x\) anymore. It is a constant function in the given interval.

Calculate the limit

Since \(f(x)\) is a constant function in the interval \((\pi - 0.01, \pi + 0.01)\), the limit is simply the value of the function: \(\lim _{x \rightarrow \pi} f(x) = f(\pi) = \frac{(\sin(9))}{3}\) Thus, the correct answer is: (c) \([(\sin 9) / 3]\)

Key concepts

Greatest Integer Function

Understanding the greatest integer function, commonly denoted by \( \lfloor x \rfloor \), is integral to unpacking our problem. This function outputs the greatest integer less than or equal to a given real number. For example, \( \lfloor 3.8 \rfloor \) returns 3 because 3 is the largest integer that does not exceed 3.8.

In our exercise, the greatest integer function plays a crucial role as it defines the intervals within which the limit is to be evaluated. It's important to note that if \(x\) approaches a whole number from either side, \( \lfloor x \rfloor \) remains constant until \(x\) crosses an integer value. In our case, as \(x\) goes to \(\pi\), \( \lfloor x \rfloor \) will continuously be 3, since \(\pi\) is approximately 3.14 and the closest integer below it is 3.

Properties of Sine Function

The sine function \( \sin(x) \) is a fundamental trigonometric function that returns the y-coordinate of a point on the unit circle corresponding to a given angle measured in radians. Some properties of the sine function crucial to this exercise include its periodic nature, range, and continuity.

One important aspect is that \( \sin(x) \) exhibits periodicity of \(2\pi\), meaning that \( \sin(x + 2\pi) = \sin(x) \) for any \(x\). The range of \( \sin(x) \) is between -1 and 1, inclusive. This boundedness ensures that the function’s output is predictable and constrained, which is helpful when calculating limits. Moreover, the sine function is continuous for all real numbers, allowing us to deal with limits smoothly. It's this continuity that permits a direct substitution in the limit calculation when \(x\) approaches a specific value.

Constant Function

A constant function is a function that returns the same value no matter what the input is. Formally, if we have a function \(f(x) = c\), where \(c\) is a constant, then for any value of \(x\), the output is always \(c\).

In the context of our limit problem, when \(x\) is in the neighborhood of \(\pi\), our function simplifies to \(f(x) = \frac{(\sin(9))}{3}\), which no longer depends on \(x\). Since it gives the same output for any input in the specified range, it is a constant function. The basic property of a constant function is that its limit, as \(x\) approaches any value, is simply the constant value itself. This characteristic makes finding the limit straightforward, as it removes the dependence on the variable \(x\). Thus, the limit of our function as \(x\) approaches \(\pi\) is the constant value \( \frac{\sin(9)}{3} \).