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

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, meaning none of the given options are correct.

Step 1: Recognize the Limit

The exercise asks us to find the limit of the expression \( \left[ \frac{e}{(1+1/x)^x} \right]^x \) as \( x \to \infty \).

Step 2: Analyze Inner Expression

First, consider the expression \( (1 + 1/x)^x \) as \( x \to \infty \). This is a well-known limit that approaches \( e \).

Step 3: Compare to e

Thus, as \( x \to \infty \), \( (1 + 1/x)^x \approx e \). Therefore, \( \frac{e}{(1 + 1/x)^x} \) approaches 1.

Step 4: Simplify the Overall Expression

Notice that each term of the expression inside the main brackets approaches 1: \( \left[\frac{e}{(1+1/x)^x}\right]^x = 1^x \). Hence, the entire expression approaches 1.

Step 5: Calculate the Limit

Finally, as \( x \to \infty \), \( 1^x \) approaches 1, so the original expression \( \lim _{x \rightarrow \infty}\left[\frac{e}{(1+1 / x)^{x}}\right]^{x} = 1 \).

Key Concepts

Exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The most widely recognized exponential function is the natural exponential function, with the base being the constant \( e \), approximately equal to 2.718. This function is written as \( e^x \), where \( x \) is the exponent.
Exponential functions have unique properties:

• Growth and Decay: Positive exponents result in growth as \( x \) increases, while negative exponents describe a decay process.
• Logarithmic Relationship: It is the inverse of a logarithmic function, reflecting the close relationship between these two concepts.
• Limits at Infinity: As demonstrated in our exercise, exponential functions play an important role in determining the behavior of sequences or functions as they approach infinity.

In the given problem, understanding the limiting behavior of exponential functions is key to simplifying complex expressions and finding their asymptotic values.

Convergence of sequences

Convergence of sequences involves determining whether the terms of a sequence approach a specific value as the index number becomes infinitely large. A sequence \( \{a_n\} \) is said to converge to a limit \( L \) if, for every positive number \( \epsilon \), there exists an index \( N \) such that for all \( n > N \), the terms satisfy \( |a_n - L| < \epsilon \).
In simpler terms, as the sequence progresses, its terms become "arbitrarily close" to the limit \( L \).

• Key Concepts:
  • Limit of a Sequence: The value a sequence approaches as its index heads to infinity.
  • Bounded and Monotonic Sequences: Convergence is guaranteed if a sequence is both bounded and monotonic (either increasing or decreasing).

In our exercise, the sequence \( (1 + 1/x)^x \) is shown to converge to \( e \) as \( x \to \infty \), a critical step in understanding the behavior of sequences involving exponential terms.

Mathematical limits

Mathematical limits describe the value a function or sequence approaches as its input or index approaches some value. Limits are foundational in calculus, providing a framework for understanding continuous functions, differentiation, and integration. They are typically expressed in the form \( \lim_{x \to c} f(x) = L \), where \( f(x) \) approaches \( L \) as \( x \) approaches \( c \).
Some essential properties of limits include:

• Uniqueness: A limit, when it exists, is unique.
• Arithmetic Operations: Limits follow specific rules for addition, multiplication, and division, making computations easier.
• Behavior at Infinity: Limits help in determining the behavior of functions as inputs approach infinity, zero, or other critical points.

In the exercise at hand, recognizing and evaluating the limit \( \lim_{x \to \infty} \left[ \frac{e}{(1 + 1/x)^x} \right]^x \) is crucial. Breaking down expressions using known limits, such as \( (1 + 1/x)^x \to e \), simplifies complex calculations and confirms the expression's convergence to a specific number at infinity.