Problem 46 - A Complete Resource Book in Mathematics for JEE Main

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 is \( \infty \) (Option D).

Step 1: Simplify the Expression

The expression given is \( \lim_{n \rightarrow \infty} \frac{\sqrt[4]{n^5+2} - \sqrt[3]{n^2+1}}{\sqrt[5]{n^4+2} - \sqrt[2]{n^3+1}} \). To analyze this, we notice the highest degrees in each term. The leading term in the numerator is \( \sqrt[4]{n^5} \) and \( \sqrt[3]{n^2} \), and in the denominator are \( \sqrt[5]{n^4} \) and \( \sqrt[2]{n^3} \). Simplifying, we reason that as \( n \) approaches infinity, constants and lower power terms become negligible.

Step 2: Factor Dominant Terms

In the numerator, factor out dominant terms: \( \sqrt[4]{n^5+2} \approx n^{5/4} \) and \( \sqrt[3]{n^2+1} \approx n^{2/3} \), hence the dominant term in the numerator is \( n^{5/4} - n^{2/3} \). In the denominator, factor as \( \sqrt[5]{n^4+2} \approx n^{4/5} \) and \( \sqrt[2]{n^3+1} \approx n^{3/2} \), giving a dominant term of \( n^{4/5} - n^{3/2} \).

Step 3: Determine the Limit Behavior

Consider the leading behavior: the largest powers will dictate the behavior as \( n \to \infty \). For the numerator, \( n^{5/4} \) and in the denominator \( n^{3/2} \). The expression approximates to \( n^{5/4 - 2/3} / n^{4/5 - 3/2} \), simplifying the powers and yielding \( n^{15/12 - 8/12} / n^{12/12 - 18/12} = n^{7/12} / n^{-6/12} = n^{1.083} \). As \( n \to \infty \), this trends to \( \infty \).

Step 4: Conclude the Limit

Since the expression leads to an overall exponent that is positive in the numerator and negative in the denominator, the solution tends to infinity as \( n \to \infty \). Therefore, the choice (D) \( \infty \) is correct.

Key Concepts

Calculus

Calculus, as a subject, dives into the study of change and motion through derivatives and integrals. A critical part of calculus is understanding limits, which help us understand the behavior of functions as they approach specific points or infinity. Limits allow us to take an "infinite" look at what happens to a function. Imagine zooming in endlessly on a graph, trying to predict how it behaves beyond visible bounds.

• Limits are foundational in finding derivatives, which measure instantaneous rates of change.
• Integration, another central concept, uses limits to sum infinitesimally small areas under curves.
• In calculus, we frequently study functions as they approach very large or very small values, helping in the analysis of asymptotic behavior.

Understanding limits is essential for grasping more complex calculus topics, from solving real-world motion problems to integrating complex functions.

Infinity

Infinity in mathematics represents an idea rather than a number. It symbolizes the concept of endlessness or boundlessness. When we talk about limits, such as in this exercise, we are frequently dealing with infinity to understand the behavior of functions as values grow exceedingly large. Infinity is not a number that can be reached, but rather a direction.
In the context of limits:

• "Asymptotic" behavior pertains to how functions behave as they approach infinity or a point.
• Working with infinity involves understanding various mathematical principles, like division by zero or multiplying large numbers.
• The limit approaching infinity in this exercise shows how the function's behavior can predict trends infinitely far along the input range.

Infinity pushes the boundary of mathematical thought, challenging us to grasp the gravitational pull of boundless space.

Asymptotic Behavior

The term "asymptotic behavior" refers to how functions behave as they approach certain lines or points infinitely closely, but never quite reach them. In calculus, understanding this behavior helps us approximate complex expressions by focusing on leading terms and significant trends.
Key points to note:

• Asymptotic analysis focuses on determining which terms dominate as variables go to infinity, allowing simplification of expressions.
• In this specific exercise, the highest power terms in numerator and denominator dictate the limit conclusion. As minor terms become insignificant at large scales, we focus on predominant terms to ascertain function trends.
• For the limit \( rac{ ext{leading terms of numerator}}{ ext{leading terms of denominator}} \), we obtain the asymptotic trend as \( n \to \infty \), concluding that it approaches infinity.

Asymptotic behavior simplifies understanding of complex expressions by reducing computations to their most impactful terms, key in predicting function limits at extreme scales.