Problem 15
Question
\(\lim _{x \rightarrow \pi / 2} \frac{\left[\frac{x}{2}\right]}{\ln (\sin x)}\) (where [.] denotes the greatest integer function) (A) Does not exist (B) equals 1 (C) equals 0 (D) equals \(-1\)
Step-by-Step Solution
Verified Answer
The limit equals 0.
1Step 1: Understand the Components
The problem involves the limit of a fraction as \(x\) approaches \(\pi/2\). The numerator is the greatest integer function (floor function) applied to \(x/2\), and the denominator is the natural logarithm of \(\sin x\). We need to evaluate this limit as \(x\) approaches \(\pi/2\).
2Step 2: Analyze the Denominator
Calculate \(\ln(\sin x)\) as \(x\) approaches \(\pi/2\). As \(x\) gets close to \(\pi/2\), \(\sin x\) approaches 1. Therefore, \(\ln(\sin x)\) approaches \(\ln(1) = 0\). This means the denominator approaches 0.
3Step 3: Analyze the Numerator
Consider the numerator, \(\left\lfloor \frac{x}{2} \right\rfloor\). When \(x\) approaches \(\pi/2\), first calculate \(\frac{x}{2}\), which approaches \(\frac{\pi}{4}\) (approximately 0.785). The greatest integer less than or equal to \(\frac{\pi}{4}\) is 0.
4Step 4: Evaluate the Limit
Now, evaluate the expression \(\lim _{x \rightarrow \pi / 2} \frac{0}{\ln(\sin x)}\). Since the numerator is 0 while the denominator tends towards 0, the entire fraction equals 0. Hence the limit value is 0.
Key Concepts
Greatest Integer FunctionNatural LogarithmTrigonometric Limits
Greatest Integer Function
The greatest integer function, often represented as \([x]\), refers to the largest integer less than or equal to a given number \(x\). It is also known by other names, like the floor function or floor notation. This function effectively "rounds down" any real number to the nearest whole integer. For instance, \([3.7] = 3\) and \([-2.3] = -3\).
To understand its application, consider the problem where \(x \to \pi/2\). Here, our task is to evaluate \(\left\lfloor \frac{x}{2} \right\rfloor\). When \(x = \pi/2\), \(\frac{x}{2}\) becomes \(\frac{\pi}{4}\), approximately 0.785. Consequently, the greatest integer less than or equal to 0.785 is 0, which simplifies our function into a manageable numerical value for further calculations.
To understand its application, consider the problem where \(x \to \pi/2\). Here, our task is to evaluate \(\left\lfloor \frac{x}{2} \right\rfloor\). When \(x = \pi/2\), \(\frac{x}{2}\) becomes \(\frac{\pi}{4}\), approximately 0.785. Consequently, the greatest integer less than or equal to 0.785 is 0, which simplifies our function into a manageable numerical value for further calculations.
Natural Logarithm
The natural logarithm, represented as \(\ln(x)\), is a logarithm with base \(e\), where \(e \approx 2.718\). It's widely used in mathematics to model exponential growth or decay and simplify the integration of exponential functions. One of its key properties is \(\ln(1) = 0\), because \(e^0 = 1\).
In our exercise, we analyze \(\ln(\sin x)\) as \(x\) approaches \(\pi/2\). As \(x\) tends towards \(\pi/2\), \(\sin x\) approaches 1. Given the property that \(\ln(1) = 0\), \(\ln(\sin x)\) thus tends towards 0 as well. This behavior of the natural logarithm in our limit calculation indicates the denominator becomes negligible near this point.
In our exercise, we analyze \(\ln(\sin x)\) as \(x\) approaches \(\pi/2\). As \(x\) tends towards \(\pi/2\), \(\sin x\) approaches 1. Given the property that \(\ln(1) = 0\), \(\ln(\sin x)\) thus tends towards 0 as well. This behavior of the natural logarithm in our limit calculation indicates the denominator becomes negligible near this point.
Trigonometric Limits
Trigonometric functions play a vital role in calculating limits, particularly when a function approaches an indeterminate form. In this exercise, we deal with \(\sin x\) as \(x\) approaches \(\pi/2\).
The value of \(\sin x\) oscillates between -1 and 1, and at specific points, such as the limits of \(0\), \(\pi/2\), and so forth, it assumes key values like 0 and 1. As \(x \to \pi/2\), \(\sin x\) tends to 1, an essential realization because it drives the behavior of the natural logarithm in the associated expression. Understanding how these classic trigonometric limits work helps simplify complex functions composed of these elements.
The value of \(\sin x\) oscillates between -1 and 1, and at specific points, such as the limits of \(0\), \(\pi/2\), and so forth, it assumes key values like 0 and 1. As \(x \to \pi/2\), \(\sin x\) tends to 1, an essential realization because it drives the behavior of the natural logarithm in the associated expression. Understanding how these classic trigonometric limits work helps simplify complex functions composed of these elements.
- \(\sin(\pi/2) = 1\)
- The limit allows simplification as \(x\) approaches \(\pi/2\), a known value facilitating computations
Other exercises in this chapter
Problem 13
$$ \begin{aligned} &\lim _{x \rightarrow 1} \frac{x \sin (x-[x])}{x-1}, \text { where }[\cdot] \text { denotes the greatest }\\\ &\text { integer function, is e
View solution Problem 14
If \(f(x)=\int \frac{2 \sin x-\sin 2 x}{x^{3}} d x, x \neq 0\), then \(\lim _{x \rightarrow 0} f(x)\) is (A) 0 (B) \(\infty\) (C) \(-1\) (D) 1
View solution Problem 16
\(\lim _{m \rightarrow \infty} \lim _{n \rightarrow \infty}\left(1+\cos ^{2 m} n ! \pi x\right)\) is equal to (A) 2 (B) 1 (C) 0 (D) None of these
View solution Problem 17
\(\lim _{x \rightarrow 0}\left[\frac{\sin ([x-3])}{[x-3]}\right]\), where \([\cdot]\) represents greatest integer function, is (A) 0 (B) 1 (C) Does not exist (D
View solution