Problem 25 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\lim _{x \rightarrow-\infty}\left[\frac{x^{4} \sin (1 / x)+x^{2}}{1+|x|^{3}}\right]\) is (A) 1 (B) \(-1\) (C) 0 (D) \(\infty\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Rewriting the Expression

First, let's rewrite the expression more clearly: \[ \lim_{x \rightarrow -\infty} \left[ \frac{x^4 \sin \left( \frac{1}{x} \right) + x^2}{1 + |x|^3} \right] \]. Here, as \(x\) approaches \(-\infty\), \(x^4\) becomes very large, whereas \(\sin\left(\frac{1}{x}\right)\) remains bounded between \(-1\) and \(1\).

Step 2: Simplifying by Dominant Terms

The dominant term in the numerator is \(x^4 \sin\left(\frac{1}{x}\right)\), but \(x^4 \sin\left(\frac{1}{x}\right)\) contributes less to the numerator compared to \(x^4\) alone because \(\sin\left(\frac{1}{x}\right)\) is always between \(-1\) and \(1\). Also, observe \( |x| = -x\) since \(x\) is negative. So, simplify as: \( \frac{x^4 \sin\left(\frac{1}{x}\right) + x^2}{1 + x^3} \).

Step 3: Approximating the Limit's Numerator and Denominator

As \(x\) approaches \(-\infty\), the most significant term in the numerator is \(x^2\) and in the denominator \(x^3\). Thus, the expression becomes approximately \( \frac{x^2}{-x^3} = -\frac{1}{x} \).

Step 4: Taking the Limit

Find the limit of \(-\frac{1}{x}\) as \(x\) approaches \(-\infty\). Since \(-\frac{1}{x}\) approaches 0 because \(\frac{1}{x}\) approaches 0, the entire limit converges to 0.

Key Concepts

Asymptotic Behavior

In mathematics, asymptotic behavior refers to how a function behaves as the input value approaches a significant point, often infinity. When dealing with limits, understanding asymptotic behavior is crucial to grasp what happens when a function stretches to large values or approaches specific input values.
Asymptotic expressions help in determining the dominant terms, thus simplifying complex expressions. For instance, in the given exercise, as the variable \( x \) approaches \(-\infty\), the expression's asymptotic behavior allows us to focus only on the dominant parts.

• The most significant impact will arise from the highest power of \( x \) in the numerator and denominator.
• This focus results in an approximation that simplifies the evaluation of the limit as \( x \) becomes very large.

Recognizing and utilizing asymptotic behavior helps us predict the function's behavior more simply, leading to a more straightforward computation of limits. Learning this concept is vital as it applies to both algebraic and more complex functions.

Infinity Approach

When we talk about the infinity approach, we refer to the limit analysis as the variable approaches positive or negative infinity. This can often mean understanding how terms grow relative to each other as the input, such as \( x \) in our expression, becomes infinite.
Limits at infinity are especially useful in understanding functions' end behavior. This exercise requires us to examine the function as \( x \) approaches \(-\infty\).

• By recognizing how the terms in our expression behave as \( x \) stretches to very large negative numbers, we can simplify the problem.
• We watch for which terms become negligible and which dominate due to their relative growth rates.

Being proficient in handling limits at infinity helps not only in theoretical calculus but also in practical applications such as physics and engineering where such conditions often model real-world behavior.

Trigonometric Limits

Trigonometric limits are a specialized area focusing on limits involving trigonometric functions. In the exercise provided, the function \( \sin\left(\frac{1}{x}\right) \) plays a role, given that \( x \) approaches \(-\infty\).
Despite \( x \) changing significantly, \( \sin\left(\frac{1}{x}\right) \) exhibits bounded behavior.

• The sine function's inherent property is its periodicity and its values always remain between \(-1\) and \(1\).
• This bounded nature influences how terms involving sine functions contribute to limits, making expressions with large coefficients like \( x^4 \sin\left(\frac{1}{x}\right) \) manageable.

Understanding trigonometric limits helps to simplify expressions where trigonometric terms do not grow unboundedly. Engaging with these concepts equips students with the ability to handle more complex problems in calculus effectively.