Problem 116 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow c} f(x)=l=\lim _{x \rightarrow c} h(x) $$ and, \(f(x) \leq g(x) \leq h(x), \forall x \in I\). Then, \(\lim _{x \rightarrow c} g(x)=l\). \(\lim _{x \rightarrow 0^{+}}\left(\lim _{n \rightarrow \infty} \frac{\left[1^{2} x^{x}\right]+\left[2^{2} x^{x}\right]+\ldots+\left[n^{2} x^{x}\right]}{n^{3}}\right)\), where \([\cdot]\) denotes the greatest integer function, is equal to (A) \(-\frac{1}{3}\) (B) \(\frac{1}{3}\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The answer is (B) \(\frac{1}{3}\).

Step 1: Understand the Nested Limits

First, we need to comprehend the question which involves two limits. The inner limit \(\lim _{n \rightarrow \infty} \frac{\left[1^{2} x^{x}\right]+\left[2^{2} x^{x}\right]+\ldots+\left[n^{2} x^{x}\right]}{n^{3}}\) needs to be evaluated first before the outer one \(\lim _{x \rightarrow 0^{+}}\).

Step 2: Analyze the Inner Limit Function

Analyze the expression inside the bracket: \(k^2 x^x\). Since \(x^x = e^{x \ln x}\), as \(x \to 0^+\), \(x \ln x \to 0\). Hence, \(x^x \to 1\). Thus each term \(\left[k^2 x^x\right]\) approaches \(k^2\) as a greatest integer.

Step 3: Simplify the Inner Limit

Thus for large \(n\), the expression approximates to \(\frac{1^2 + 2^2 + \ldots + n^2}{n^3}\). The sum \(1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}\).

Step 4: Evaluate the Inner Limit as \(n \to \infty\)

Now simplify: \(\frac{\frac{n(n+1)(2n+1)}{6}}{n^3} = \frac{(n(n+1)(2n+1))}{6n^3} = \frac{(2n^3 + 3n^2 + n)}{6n^3} = \frac{1}{3} + O(\frac{1}{n})\). As \(n \to \infty\), this limit is \(\frac{1}{3}\).

Step 5: Evaluate the Outer Limit as \(x \to 0^{+}\)

Given \(\lim _{n \rightarrow \infty}\) is constant \(\frac{1}{3}\), the outer limit \(\lim_{x \to 0^+} \frac{1}{3}\) is simply \(\frac{1}{3}\), since it's a constant function independent of \(x\).

Key Concepts

Limits

The concept of limits is a fundamental idea in calculus. It describes the behavior of a function as its input approaches a certain value. This allows us to understand what value a function could potentially reach, even if it never actually does so. In problems like the one provided, where you encounter nested limits, it’s essential to work from the inside out.

• Start by understanding the inner limit. You first evaluate what the function does when you approach infinity or some other limit point inside the nested structure.
• Once the inner limit is solved, proceed to the outer limit, considering how the result of the previous step behaves as you approach the new limit point.

This approach can simplify complex problems, making them more manageable.

Greatest Integer Function

The greatest integer function, denoted by brackets \([\cdot]\), returns the largest integer less than or equal to a given number. This function is useful in various scenarios where rounding down to an integer is required. For instance, in this problem, you apply this function to terms like \(k^2x^x\) as \(k\) iterates through integers.

Why Use It?

• It helps in simplifying terms by reducing them to integer values, which are often easier to handle.
• This function plays a key role in the evaluation of the limit by impacting the sum and how each term contributes as \(n\) grows.

Understanding how greatest integer approximation affects each term in a series is crucial for solving limit problems involving sums.

Real Valued Functions

Real valued functions are functions that take real numbers as input and return real numbers as output. These types of functions are common in calculus, where understanding their behavior as variables change is key.

• They often express relationships between variables in mathematical models.
• In optimization, limits, and calculus, real valued functions are crucial for modeling real-world phenomena.

In the context of your problem, functions \(f, g,\) and \(h\) are all real valued, representing mathematical expressions applicable over a real interval. Analyzing these functions using tools like the limit and greatest integer function allows us to understand their behavior at specific points, like as \(x\) approaches 0.