Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{n \rightarrow \infty} \frac{\sqrt[4]{n^{5}+2}-\sqrt[3]{n^{2}+1}}{\sqrt[5]{n^{4}+2}-\sqrt[2]{n^{3}+1}}\) is (A) 1 (B) 0 (C) \(-1\) (D) \(\infty\)

Step-by-Step Solution

Answer

The limit evaluates to 0; option (B) is correct.

Step 1: Analyze the Limit Expression

We are given a complex limit expression involving radical expressions. The problem requires us to evaluate \[ \lim_{n \to \infty} \frac{\sqrt[4]{n^5 + 2} - \sqrt[3]{n^2 + 1}}{\sqrt[5]{n^4 + 2} - \sqrt[2]{n^3 + 1}}. \] To solve this, we must understand how each term behaves as \( n \to \infty \).

Step 2: Simplify the Numerator

The numerator is \( \sqrt[4]{n^5 + 2} - \sqrt[3]{n^2 + 1} \). As \( n \to \infty \), \( \sqrt[4]{n^5 + 2} \approx \sqrt[4]{n^5} = n^{5/4} \) and \( \sqrt[3]{n^2 + 1} \approx \sqrt[3]{n^2} = n^{2/3} \). Thus, the dominant term is \( n^{5/4} \), which simplifies the expression to \( n^{5/4} - n^{2/3} \).

Step 3: Simplify the Denominator

The denominator is \( \sqrt[5]{n^4 + 2} - \sqrt[2]{n^3 + 1} \). As \( n \to \infty \), \( \sqrt[5]{n^4 + 2} \approx \sqrt[5]{n^4} = n^{4/5} \) and \( \sqrt[2]{n^3 + 1} \approx \sqrt[2]{n^3} = n^{3/2} \). Thus, the dominant term is \( n^{3/2} \), which simplifies the expression to \( n^{4/5} - n^{3/2} \).

Step 4: Evaluate the Limit of Rational Expression

We are now examining the expression \[ \lim_{n \to \infty} \frac{n^{5/4} - n^{2/3}}{n^{4/5} - n^{3/2}}. \] As \( n \to \infty \), only the highest power terms are significant: \( n^{5/4} \) and \( -n^{3/2} \), since these dominate the other terms. The expression simplifies to \[ \lim_{n \to \infty} \frac{n^{5/4}}{-n^{3/2}} = \lim_{n \to \infty} -n^{5/4 - 3/2} = \lim_{n \to \infty} -n^{-1/4}. \]

Step 5: Evaluate the Final Limit

Since \(-n^{-1/4}\) approaches 0 as \( n \to \infty \), the final limit of the original expression is 0. Thus, \[ \lim_{n \to \infty} \frac{\sqrt[4]{n^5 + 2} - \sqrt[3]{n^2 + 1}}{\sqrt[5]{n^4 + 2} - \sqrt[2]{n^3 + 1}} = 0. \]

Key Concepts

Infinity

Infinity represents a concept that goes beyond any numerical value we know. It's not a number but an idea of something that grows endlessly large. In calculus, particularly while examining limits, students encounter infinity when they consider the behavior of functions as variables grow indefinitely.

Here's an insight into infinity with limits:

• When we say \( n \to \infty \), we're observing what happens to a function as \( n \) increases without bound.
• In a rational expression involving infinity, like our exercise, the challenge is determining which terms in the expression dominate as \( n \to \infty \).

Recognizing concepts around infinity helps us see whether a function stabilizes, grows, or declines as it moves toward infinite values.

Dominant Term

In rational expressions or any mathematical expressions with multiple terms, the dominant term is the part that has the most significant impact on the value of the expression as it evolves, especially as variables move towards infinity.

To identify dominant terms:

• Focus on the terms with the highest power of the variable, since they have the most influence as variables become large.
• In the provided exercise, for the numerator \( n^{5/4} \) is the dominant term as \( n \to \infty \), compared to \( n^{2/3} \).
• Similarly, in the denominator, \( -n^{3/2} \) is dominant compared to \( n^{4/5} \).

Recognizing these dominant terms allows us to simplify and approximate expressions for easier analysis, especially when dealing with limits.

Rational Expression

A rational expression is a fraction where both the numerator and the denominator are polynomials or algebraic expressions. They can get quite complex, especially involving radicals or higher powers, as seen in our problem.

Key points about rational expressions:

• Both parts must be considered to understand the behavior of the whole expression.
• They can be simplified by examining the dominant terms, especially when evaluating their limits.
• In our exercise, the rational expression changes dramatically as \( n \to \infty \) because of the terms \( n^{5/4} \) and \( -n^{3/2} \).

Approaching rational expressions with this analysis is instrumental in solving limits, simplifying complex fractions, and understanding their long-term behaviors.

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as square roots, cube roots, and higher-order roots. They often appear in limit problems where their properties influence the function's behavior significantly.

Considerations for radical expressions:

• Simplifying radical expressions involves identifying dominant terms just as we do with rational expressions.
• Each radical's behavior needs to be understood separately before understanding their combined impact, especially in limits.
• In the limit problem provided, simplifying \( \sqrt[4]{n^5 + 2} \approx n^{5/4} \) illustrates how a radical's effect is distilled to its dominant power of \( n \).

These steps support breaking down complex radical expressions into more manageable forms for analysis in calculus, especially when dealing with concepts like infinity.