Problem 141 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{x \rightarrow \infty}\left(1-\frac{1}{x^{2}}\right)^{x} \text { \\{Ans. } 1\right\\} $$

Step-by-Step Solution

Answer

The short answer is: \(\lim_{x \rightarrow \infty}\left(1-\frac{1}{x^{2}}\right)^{x} = \frac{1}{e} \approx 0.368\).

Step 1: Substitute the variable

Firstly, let's substitute the variable \(y=\frac{1}{x^2}\), then we have \(x^2 = \frac{1}{y}\) or \(\frac{1}{x^2} = y\). As \(x\) approaches to infinity, \(y\) approaches to 0. So, we can rewrite the given limit as follows: \(\lim _{x \rightarrow \infty}\left(1-\frac{1}{x^{2}}\right)^{x} = \lim _{y \rightarrow 0}\left(1-y\right)^{\frac{1}{y}}\)

Step 2: Rewrite and apply the well-known limit

Now, let's consider \(a=-1\) in the well-known limit \(\lim_{k \to \infty} \left(1 + \frac{a}{k}\right)^k = e^a\). So, we have: \(\lim_{k \to \infty} \left(1 - \frac{1}{k}\right)^k = e^{-1}\) Now, compare this expression with the given expression of our problem: \(\lim _{y \rightarrow 0}\left(1-y\right)^{\frac{1}{y}}\) These two limits are equivalent with k can be replaced by \(\frac{1}{y}\). Thus, we can apply the well-known limit to our given limit.

Step 3: Find the limit

Now that we know the relationship between the limit in our problem and the well-known limit. We plug in the value of \(a\), \(\lim_{y \to 0} \left(1-y\right)^{\frac{1}{y}} = e^{-1}\) By calculating the value of \(e^{-1}\), we have: \(e^{-1} = \frac{1}{e} \approx 0.368\) Hence, the limit of the given function as \(x\) approaches infinity is: \(\lim _{x \rightarrow \infty}\left(1-\frac{1}{x^{2}}\right)^{x} = \frac{1}{e} \approx 0.368\)

Key Concepts

Convergence and Divergence

In calculus, the concepts of convergence and divergence refer specifically to the behavior of sequences and functions as they approach a particular point or infinity. Convergence means that a sequence or function approaches a specific, finite limit. In contrast, divergence implies that it does not approach a particular value but instead heads off to infinity, or behaves erratically without approaching any value.

By analyzing the convergence or divergence of a function, particularly those involving a limit, we gain critical insight into its long-term behavior. For example, in the textbook exercise, we see the function \(\left(1-\frac{1}{x^{2}}\right)^{x}\) as \(x\) approaches infinity. Through a process of substitution and comparison with a well-known limit, we determine that as \(x\) gets larger and larger, the function converges to \(\frac{1}{e}\), a clear indication of its behavior as \(x\) approaches infinity.

Exponential Functions

Exponential functions are a significant area in calculus characterized by their unique property of having a constant rate of growth or decay. The general form of an exponential function is \(f(x) = a^x\), where \('a'\) is a constant base and \('x'\) is the exponent. \These functions are fascinating due to their behavior at extremes; as \(x\) approaches infinity, an exponential function will either grow without bound or approach zero, depending on the base value. In the exercise, we have a variant of an exponential function where the base, \(1-\frac{1}{x^{2}}\), is slightly less than 1. As \(x\) approaches infinity, that base approaches 1 which is why the function has a finite limit—this showcases a unique case where the exponential form converges rather than diverges or grows without bound.

Infinite Limits

When we talk about infinite limits, we're exploring the behavior of functions as they grow without bound in either the positive or negative direction. It's essentially observing what happens to a function as we plug in larger and larger values for \(x\), or as \(x\) approaches a critical number that makes the function spike towards infinity.

However, not all functions with 'infinity' in play are headed towards boundlessness. Some, like the function in our textbook problem, \(\left(1-\frac{1}{x^{2}}\right)^{x}\) as \(x\) approaches infinity, tend to a finite limit, which in this case is \(\frac{1}{e}\). This demonstrates that even when dealing with limits involving infinity, the outcomes can be diverse, and understanding the specific characteristics of the function is crucial.

Properties of Limits

Discussing the properties of limits is essential for solving limit problems efficiently and accurately. These properties are rules that allow us to manipulate limits in algebraic ways to make them easier to solve. Some fundamental properties include the limit of a sum being the sum of the limits, the limit of a product being the product of the limits, and the power rule which allows us to bring exponents outside the limit.

Applying Well-Known Limits

Especially relevant in our exercise is the application of well-known limits. By recognizing the expression \(\left(1-\frac{1}{x^{2}}\right)^{x}\) as akin to the exponential function's limit that gives Euler's number \(e\), we were able to directly apply a known result to find that the limit is \(\frac{1}{e}\). Such manipulations underscore the importance of understanding the properties of limits in simplifying complex-looking problems into familiar forms.