Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the fact that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f\left(\frac{1}{x}\right)\) to find the limits. $$ \lim _{n \rightarrow \infty}\left(\frac{n-1}{n+1}\right)^{n} $$

Step-by-Step Solution

Answer

The limit is \( e^{-2} \).

Step 1: Understanding the Limit Transformation

To find \( \lim_{n \rightarrow \infty}\left(\frac{n-1}{n+1}\right)^n \), we first rewrite \( \left(\frac{n-1}{n+1}\right)^n \) using the transformation given in the problem: \( \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow 0^+} f\left(\frac{1}{x}\right) \). This indicates a transformation involving a change from large \( n \) to small values through inversion.

Step 2: Simplifying the Expression

Observe that \( \frac{n-1}{n+1} \approx 1 - \frac{2}{n+1} \) as \( n \) becomes large. Thus, we have the base \( 1 - \frac{2}{n+1} \). We will be evaluating \( \left(1 - \frac{2}{n+1}\right)^n \).

Step 3: Using the Exponential Limit Approximation

For very small \( x \), \( (1 - x)^n \) can be approximated using the limit \( \ln(1-x) \approx -x \) for small \( x \). Therefore, we have: \((1-x)^n \approx e^{-nx}\). Substitute \( x = \frac{2}{n+1} \): \(\left(1-\frac{2}{n+1}\right)^n \approx e^{-n \cdot \frac{2}{n+1}}.\) As \( n \rightarrow \infty \), it simplifies to: \( e^{-2}.\)

Step 4: Evaluate the Limit

Therefore, the original limit \( \lim_{n \rightarrow \infty}\left(\frac{n-1}{n+1}\right)^n \) is equivalent to evaluating: \( \lim_{n \rightarrow \infty} e^{-2} = e^{-2}. \) As there are no variables left in the expression, this is simply the value \( e^{-2}. \)

Key Concepts

Limit Transformation

In calculus, transforming limits is a valuable tool, especially when dealing with expressions that progress towards infinity. One key method involves utilizing the identity \( \lim_{x \to \infty} f(x) = \lim_{x \to 0^+} f\left(\frac{1}{x}\right) \). This transformation switches the behavior of a function at infinity to its behavior as the variable approaches zero. This allows us to handle complex limits by transforming them into more manageable forms.

• Limit transformations are particularly helpful for studying the asymptotic behavior of functions.
• This approach works by essentially flipping the variable, providing new insights by studying the limit from a different perspective.

In our example, such a limit transformation facilitated the expression \(\left(\frac{n-1}{n+1}\right)^n\) by turning the problem into one that examines small perturbations around one. This strategy makes the limit evaluation more straightforward as it closely ties into known approximations.

Exponential Limit Approximation

The exponential limit approximation is a powerful technique when dealing with limits of expressions raised to a power that tend towards zero. When a term like \((1-x)^n\) is present and \(x\) is small, the expression can be approximated with exponentials using the natural logarithm: \( \ln(1-x) \approx -x \) for small \(x\). This allows us to use the famous approximation: \((1-x)^n \approx e^{-nx}\).

Using this approximation makes complex limits easy to solve, especially when the terms converge rapidly due to exponential functions. In our discussion, substituting \(x = \frac{2}{n+1}\) and applying this approximation as \(n\) grows large results in \( e^{-n \cdot \frac{2}{n+1}} \). As \(n\) approaches infinity, this further simplifies to \(e^{-2}\).

• This technique is useful for evaluating limits involving repeated multiplication of similar terms.
• Exponential approximations provide smooth approaches to convergence, often resulting in strikingly simple expressions.

Limit Evaluation Techniques

Effective evaluation of limits often requires employing a mixture of analytical strategies. Techniques such as approximation, transformations, and the recognition of known patterns are pivotal in simplifying expressions that might initially appear daunting. Here’s a summary of tactics:

• Simplification: Reduce the complexity of the expression; this might involve factoring or basic algebraic manipulation, such as recognizing that \(\frac{n-1}{n+1} \approx 1 - \frac{2}{n+1}\) for large \(n\).
• Approximation: Use limit approximations—such as treating small perturbations with exponential or logarithmic approximations—to streamline calculations as we did with exponential limits.
• Transformation: Utilize transformations to change the form of the limit and analyze it from alternate perspectives that might be easier, as seen when converting behavior at infinity to behavior near zero.

Each technique depends on discerning the characteristics of the limit problem at hand. For instance, in our process, by synthesizing these techniques, we derived the result efficiently as \(e^{-2}\), showcasing the effectiveness of blending techniques to achieve accurate solutions.