Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{n \rightarrow \infty}\left[\sqrt[3]{n^{2}-n^{3}}+n\right]\) is (A) \(\frac{1}{3}\) (B) \(-\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(-\frac{2}{3}\)

Step-by-Step Solution

Answer

The limit is -1, which does not match any given option. Re-evaluate exercise context.

Step 1: Expression Rewrite

Rewrite the expression inside the limit: \( f(n) = \sqrt[3]{n^2 - n^3} + n \). Recognize that \( n^2 - n^3 = n^3 ( \frac{1}{n} - 1) \). This will help in breaking down the cubic root.

Step 2: Simplify Inside the Cubic Root

Simplify further by considering the expression inside the cubic root, \( \sqrt[3]{n^3 (\frac{1}{n} - 1)} = n (\frac{1}{n} - 1)^{1/3} \).Now, reexamine \( f(n) = n ((1 - \frac{1}{n})^{1/3}) + n \).

Step 3: Evaluate the Limit

As \( n \to \infty \), notice that \( \frac{1}{n} \to 0 \), so \( 1 - \frac{1}{n} \to 1 \). Thus, \((1 - \frac{1}{n})^{1/3} \to 1^{1/3} = 1\). Therefore, \( f(n) \approx n(1) + n = n + n = 2n\).

Step 4: Apply the Limit

Now evaluate the limit \( \lim_{n \to \infty} (n + n) = \lim_{n \to \infty} 2n \).As \( n \to \infty \), this tends to \( \infty \), so reevaluate the initial simplification if the question seeks an incorrect conclusion.

Step 5: Correct Evaluation

Re-assess, thinking carefully about dominant terms. The term with the highest growth rate for large \(n\) is \( -n^3 \). Thus,\(\sqrt[3]{n^2 - n^3} \approx \sqrt[3]{- n^3} = -n \). Combining \( f(n) = -n + n = 0 \), reaffirm correct evaluation.Thus, limit converges contrary to previous misstep due to misinterpretation of algebraic initial steps.

Key Concepts

Cubic Root Simplification

Simplifying cube roots can help us solve complex expressions. When you see \(\sqrt[3]{n^2 - n^3}\), it might initially appear perplexing. However, we can break it into more manageable parts. We notice that \(n^2 - n^3\) can be rewritten as \(n^3 (\frac{1}{n} - 1)\). This factorization makes it easier to apply the cube root.
Using the property \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), the expression inside the cube root simplifies to \(n(\frac{1}{n} - 1)^{1/3}\). This shows how algebraic manipulation and recognizing patterns in factorization can simplify the original expression.
This simplification is crucial: it helps identify the dominating behavior of the function as \(n\) becomes very large.

Dominant Term Analysis

When working with limits, especially as \(n\) approaches infinity, identifying the dominant term becomes essential. The dominant term is the part of the function that grows the fastest as \(n\) increases.
In our expression, \(n^2 - n^3\), the dominant term is \(-n^3\), as it has the largest exponent. As \(n\) grows, this term will substantially outweigh the others.
Recognizing this allows us to approximate the cube root expression as \(-n\). By focusing on \(-n^3\) in the cube root, we realize the simplified form is \(-n\), which helps balance the rest of the limit evaluation.
Understanding dominant terms aids not just in simplifying expressions, but also in predicting the behavior of functions for very large values of \(n\).

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using algebraic rules. It's a powerful tool that helps in solving complex functions easily.
In the given problem, initially, we have \(f(n) = \sqrt[3]{n^2 - n^3} + n\). By recognizing that \(n^3\) terms dominate, we rewrite \(n^2 - n^3\) as \(n^3(\frac{1}{n} - 1)\). This is an instance of algebraic manipulation.

• Factorization helps recognize key terms.
• New representation eases implementing the cube root.

As we simplify, ensuring each step is clear is crucial, making algebra a valuable skill for deciphering limits and infinite scenarios.

Infinite Limits

Infinite limits occur when the value of a function approaches infinity as \(n\) approaches an unlimited boundary. They can initially seem daunting, but understanding the behavior of each term can simplify the process.
In this exercise, by finding the dominant term, we learned that as \(n\) approaches infinity, the overall expression simplifies to \(-n + n\). Hence, the function tends towards zero: a crucial re-evaluation point.

• Identifying the behavior of large terms helps predict function outcomes.
• Overlooking negative signs and correctly applying root transformations can yield correct simplifications.

By staying vigilant about the behavior at infinite boundaries, which involves understanding how large terms influence outcomes, mastering infinite limits becomes more achievable.