Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{x \rightarrow \infty}\left[\frac{e}{(1+1 / x)^{x}}\right]^{x}=\) (A) \(e\) (B) \(e^{-1}\) (C) \(e^{1 / 2}\) (D) \(e^{-1 / 2}\)

Step-by-Step Solution

Answer

The limit is 1, matching the value of \( e^0 \).

Step 1: Understanding the Limit Expression

The given expression is \( \lim_{x \to \infty} \left[ \frac{e}{(1 + 1/x)^x} \right]^x \). Our goal is to find the limit of this expression as \( x \to \infty \). We note that \( (1 + 1/x)^x \) is a well-known expression that approaches \( e \) as \( x \to \infty \).

Step 2: Simplifying the Base

Substitute the known limit \( \lim_{x \to \infty} (1 + 1/x)^x = e \) into the expression. The base of the power becomes \( \frac{e}{e} = 1 \). Hence, the expression simplifies to \( \left[ 1 \right]^x \).

Step 3: Evaluating the Limit

Since \( 1^x = 1 \) for any \( x \), we take the limit as \( x \to \infty \). Thus \( \lim_{x \to \infty} 1^x = 1 \).

Step 4: Compare with Options

The final result of the limit is 1. Compare this with the given options. None of the provided options match exactly because they are powers of \( e \). Recognize that \( e^0 = 1 \). Hence, the matching choice with \( e^0 \) is not explicitly provided, but can be derived from (B) by understanding exponential notation.

Key Concepts

Limit evaluation

Evaluating limits, especially at infinity, helps us understand the behavior of functions as variables approach large values. Limit evaluation involves analyzing expressions to determine their ultimate value. In this exercise, we are tasked with finding the limit of an expression as \( x \) approaches infinity. The original expression is \( \lim_{x \to \infty} \left[ \frac{e}{(1 + 1/x)^x} \right]^x \). The first step in evaluating this limit is to simplify the expression. We recognize that \((1 + 1/x)^x\) is a well-known limit that approaches \( e \) as \( x \to \infty \). This allows us to substitute \( e \) into the expression, simplifying it further to \( \left[ \frac{e}{e} \right]^x = [1]^x \).
Taking the limit of \([1]^x\) as \(x\) approaches infinity is straightforward. Since the base is 1, \( [1]^x = 1 \) for any value of \( x \). Hence, the limit as \( x \to \infty \) of this expression remains 1. Ultimately, comparing the result with the available options shows none match 1 directly. However, we understand that \( e^0 = 1 \), implying the limit corresponds indirectly to the expression \( e^0 \). This exercise underscores the importance of recognizing known limits and carefully manipulating expressions to evaluate limits effectively.

Exponential functions

Exponential functions are fundamental in mathematics and real-world applications, characterized by a constant raised to the power of a variable. These functions grow rapidly and have unique properties that make them interesting. In the context of this problem, an exponential expression appears in the form \( (1 + 1/x)^x \). This expression converges to the exponential constant \( e \) as \(x\) approaches infinity.

• The base of the natural logarithm, \( e \), is approximately 2.718 and arises in countless mathematical scenarios, particularly in calculus.
• Exponential functions exhibit continuous growth or decay rates, making them crucial in modeling exponential growth in populations, compound interest, and more.
• Understanding the behavior of expressions like \( (1 + 1/x)^x \) is pivotal in grasping concepts of limits and their connection to exponential functions.

In this exercise, we simplified the original expression by recognizing the convergence to \( e \), transforming it into a more manageable form for limit evaluation. By reducing complex exponential expressions to basics, we can accurately determine limits.

Infinity limit problem

Infinity limit problems involve evaluating functions as variables approach extreme values, often requiring simplification or known limits. Solving these problems can reveal insights into asymptotic behavior or grow trends of functions at infinity. In this exercise, we faced an infinity limit problem: \( \lim_{x \to \infty} \left[ \frac{e}{(1 + 1/x)^x} \right]^x \).
Recognizing that \( (1 + 1/x)^x \to e \) allowed us to simplify the limit problem to \( [1]^x \), thus easing the evaluation task. Here are some key observations about infinity limits and their solutions:

• Infinity limit problems often necessitate simplification by reducing expressions to elemental counterparts or using established limits.
• The process often involves recognizing indeterminate forms and resolving them through algebraic manipulation or known limit properties.
• Effective handling of infinity limits involves conceptual understanding and sometimes creative approaches to manipulate expressions.

In conclusion, tackling infinity limit problems systematically through substitution and simplification exposes the underlying structure of complex functions, making the seemingly daunting task of evaluating them at infinity feasible and intuitive.