Problem 116 - A Complete Resource Book in Mathematics for JEE Main 2017

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 limit evaluates to 0, which corresponds to option (C).

Step 1: Understanding the Problem

We need to find the solution to the limit problem: \(\lim _{x \rightarrow 0^{+}}\left(\lim _{n \rightarrow \infty} \frac{\left[1^{2} x^{x}\right]+\left[2^{2} x^{x}\right]+\ldots+\left[n^{2} x^{x}\right]}{n^{3}}\right)\). Here, \([\cdot]\) denotes the greatest integer function.

Step 2: Analyzing the Inner Limit

First, let's analyze the expression inside the limit as \(n \rightarrow \infty\). The expression \(k^2 x^x\), where \(x\rightarrow 0^+\), tends to 0 because \(x^x = e^{x \ln x} \rightarrow 1\) as \(x \to 0^+\). Consequently, \([k^2 x^x] = 0\) for small \(x\).

Step 3: Simplifying the Sum for Large n

The sum \([1^2 x^x]+[2^2 x^x]+\ldots+[n^2 x^x]\) becomes zero for sufficiently small \(x\), irrespective of \(n\), since each term \([k^2 x^x] = 0\). Therefore, the numerator of the fraction remains zero.

Step 4: Evaluating the Inner Limit

Since the numerator is zero, the expression \(\frac{\left[1^{2} x^{x}\right]+\left[2^{2} x^{x}\right]+\ldots+\left[n^{2} x^{x}\right]}{n^3}\) simplifies to 0 as \(n \rightarrow \infty\). So, \(\lim_{n \rightarrow \infty} \frac{\text{numerator}}{n^3} = 0\) at each \(x\).

Step 5: Taking the Outer Limit

Now we evaluate \(\lim_{x \rightarrow 0^+} 0 = 0\). Since the inner limit evaluates to 0 for every \(x\), the outer limit, which is also 0, remains unaffected.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, often denoted as \([x]\), is a fundamental concept in mathematics that assigns to any real number the greatest integer less than or equal to that number. Imagine it as a "floor" function that rounds down to the nearest whole number. For instance, for the value 3.7, the greatest integer function would result in 3. Similarly, \([2.9]\) would also be 2. It's important not to confuse this with the ceiling function, which rounds up to the nearest integer.

This function plays a significant role in many mathematical problems, especially in limit problems where specific behavior occurs as variables approach certain values. For the exercise at hand, the function \([k^2 x^x]\) implies that for very small values of \(x\), the term becomes zero since \(x^x\) tends very close to zero as \(x o 0^+\). Thus, every term inside the brackets rounds down to zero, simplifying many complex limit expressions into manageable forms.

Squeeze Theorem

The Squeeze Theorem is an elegant theorem useful for evaluating limits, especially when direct computation is challenging. It essentially states that if a function \(g(x)\) is "squeezed" between two other functions \(f(x)\) and \(h(x)\) which have the same limits as \(x\) approaches a certain point, then the limit of \(g(x)\) must also equal that same value. In formal terms, if \( ext{lim}_{x \to c} f(x) = ext{lim}_{x \to c} h(x) = L\) and \(f(x) \leq g(x) \leq h(x)\) for all \(x\) in some interval around \(c\), except possibly at \(c\), then \( ext{lim}_{x \to c} g(x) = L\).

In an intuitive way, picture it like a sandwich where the middle piece ( ext{\

Squeeze Theorem

The Squeeze Theorem is an elegant theorem useful for evaluating limits, especially when direct computation is challenging. It essentially states that if a function \(g(x)\) is \

Nested Limits

Nested limits involve finding the limits of functions where one limit process is embedded inside another. This can reveal how a function behaves under compound limiting conditions. Consider the exercise: \(\lim_{x \to 0^+}\left(\lim_{n \to \infty} \frac{\left[1^2 x^x\right]+\left[2^2 x^x\right]+\ldots+\left[n^2 x^x\right]}{n^3}\right)\). There are two limits at play here: an "inner" limit as \(n \) approaches infinity and an "outer" limit as \(x \) approaches zero from the positive side.

It's crucial to approach these problems systematically, beginning with the inner limit. By evaluating the behavior of the expression as \(n \) becomes very large, the inner limit reveals its value. After resolving the inner limit, the result is then used as input for the outer limit. In our scenario, because the inner limit simplifies to zero for each value of \(x\), the outer limit is applied to zero, giving the final result of zero. Understanding the interplay of nested limits is fundamental in analyzing how complex expressions behave as variables change together.