Problem 272 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{\frac{x+1}{x}} \text { \\{Ans. } 1\right\\} $$

Step-by-Step Solution

Answer

The short answer to the problem is: \(\lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{\frac{x+1}{x}} = 1\).

Step 1: Substitute the limit into the expression

First, let's substitute \(x\) approaching infinity into our expression: \[ \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{\frac{x+1}{x}} \]

Step 2: Simplify the exponent

Now, let's simplify the exponent by dividing the numerator and denominator by \(x\): \[ \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{\frac{1+1/x}{1}} \] Since \(x\) is going to infinity, the term \(\frac{1}{x}\) will go to zero, making the exponent equal to 1: \[ \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{1} \]

Step 3: Find the limit

Now we can calculate the limit as \(x\) approaches infinity: \[ \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right) = 1 + \lim_{x \rightarrow \infty}\frac{1}{x} \] As \(x\) goes to infinity, \(\frac{1}{x}\) goes to zero, and the limit becomes: \[ 1 + 0 = 1 \] So, the limit of the given expression is 1, which is our final answer.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field forms the basis for much of modern mathematics and is instrumental in many areas of science and engineering. Studying calculus provides the tools to understand changes between values that are related by a function. When calculus tackles topics such as limits at infinity, it helps us understand the behavior of functions as they approach values that are not finite. Integral calculus and differential calculus are the two main divisions of calculus; the former concerning accumulation of quantities and the latter focusing on rates of change.

Infinity in Limits

The concept of infinity in limits is fascinating because it gives us insight into the behavior of functions as they approach larger and larger values. When we say that a limit is taken as 'x' approaches infinity, we're essentially considering the end behavior of a function. This can help us determine how it will act in extreme cases. Understanding limits at infinity is crucial in fields such as mathematical analysis and physics, where such conditions regularly occur. When we evaluate these limits, we often see the function either approaching a specific value or growing without bound. However, the manner in which we approach infinity—a concept without an actual 'value'—requires careful mathematical definition and consideration.

Limit Properties

Limit properties are the rules that allow us to compute limits of functions easily and methodically. These properties include the limit of a sum, the limit of a product, and the limit of a quotient. For example, the limit of a sum property states that the limit of a sum is equal to the sum of the limits, provided that these limits exist. When dealing with limits at infinity, these properties become particularly useful. They can often simplify complex expressions into more manageable pieces that can be evaluated separately, as seen in the exercise. Knowing when and how to apply these properties is a fundamental skill in the study of calculus.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, sequences, and series. It is essentially an extension of calculus and provides a rigorous foundation for calculus concepts, such as limits at infinity. Analysis is concerned not just with the computational aspects of these concepts but also with proving the theoretical underpinnings and relationships between them. It provides tools for quantifying and understanding precision, convergence, and continuity, making it indispensable for both theoretical and applied mathematics.