Problem 47 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}-\frac{4}{x^{2}}\right)^{2 x}=e^{3}\), then 'a' is equal to : [Online April 9, 2016] (a) 2 (b) \(\frac{3}{2}\) (c) \(\frac{1}{2}\) (d) \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The value of 'a' is \(\frac{3}{2}\).

Step 1: Identify the known forms

Recall that the expression \( \lim_{x \to \infty} \left(1 + \frac{b}{x} \right)^{cx} = e^d \) corresponds to the exponential limit property when \(b=c=d\). We will use this form to recognize and solve for \(a\) in the given problem.

Step 2: Rewrite the problem expression

Write the expression: \[\lim_{x \to \infty} \left(1 + \frac{a}{x} - \frac{4}{x^2} \right)^{2x} = e^3.\]Notice that the term \(-\frac{4}{x^2}\) becomes negligible as \(x\) approaches infinity.

Step 3: Simplify the expression

Assume that the term \(-\frac{4}{x^2}\) is negligible, simplifying the expression to:\[\lim_{x \to \infty} \left(1 + \frac{a}{x} \right)^{2x} = e^3.\]In this simplified form, compare it to \( \lim_{x \to \infty} \left(1 + \frac{b}{x} \right)^{cx} = e^d\).

Step 4: Equate the parameters

Given the basic form \( \lim_{x \to \infty} \left(1 + \frac{b}{x} \right)^{cx} = e^d \) where \(b=c=d\), we find that \(a = \frac{3}{2}\) since \(2a = 3\).

Step 5: Verify the result

Re-check that the relationship \(2a = 3\) is maintained, ensuring that with \(a=\frac{3}{2}\), the expressions on both sides of the equation are consistent.

Key Concepts

Exponential Limit Property

The Exponential Limit Property is a critical concept in calculus, particularly when dealing with limits involving expressions raised to a power as the variable approaches infinity. This property allows us to simplify complex expressions into a more manageable form. In mathematical terms, it often shapes expressions close to

• \(\lim_{x \to \infty} \left(1 + \frac{b}{x} \right)^{cx} = e^d \)
where \(b = c = d\).

This can be used to convert certain limit expressions into exponential form. As seen in our problem, the goal is to match the expression to this classic structure so we can use known exponential limit properties to find unknown variables. Here, the approximation simplifies our problem to find \(a\) with knowledge that
\(\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^{2x} = e^3\), thus solving \(2a = 3\) hence \(a = \frac{3}{2}\). This property is especially useful when other terms fade away as the limit tends toward infinity.

Indeterminate Forms

Indeterminate forms arise in calculus when evaluating limits and the outcome is initially unclear. They include classic situations like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), leading to ambiguous results without further manipulation.

In the given exercise, though not directly forming an indeterminate form, our expression initially contains competing terms, \(\frac{a}{x}\) and \(\frac{4}{x^2}\), that seem to compete against each other as \(x\) approaches infinity. The term \(-\frac{4}{x^2}\) becomes negligible due to its more rapid decay compared to \(\frac{a}{x}\). Thus, we can focus on simplifying the dominant term using known limit properties without resulting in an indeterminate form. The correct adjustment helps avoid unnecessary complexity and directs our solution correctly using limits analytical strategies.

Asymptotic Analysis

Asymptotic analysis is fundamental for understanding the behavior of functions as variables approach certain limits, such as infinity. Practically, it involves identifying dominant terms in an expression and predicting behavior as one or more variables grow large.

• It helps simplify complex mathematical problems by focusing on dominant behavior.
• Used in calculating limits and can guide solutions towards expected results.

In the example problem, asymptotic analysis supports the decision to neglect \(-\frac{4}{x^2}\) as \(x\) approaches infinity. By focusing only on the significant contributing term, \(\frac{a}{x}\), we apply asymptotic behavior, taking the limit where only the components that significantly impact the outcome are considered.

In broader perspectives, this type of analysis streamlines problem-solving by targeting lasting trends rather than temporary fluctuations. Recognizing negligible terms in limit problems allows seamless integration of exponential limit properties in solving for critical components strongly influencing results like the value of \(a\) in the exercise solution at hand.