Problem 39 - Essential Calculus Early Transcendentals

Question

\(39-40=\) Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value. $$\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}$$

Step-by-Step Solution

Answer

The limit \( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x = e^2 \approx 7.389 \).

Step 1: Graphical Estimation

To estimate the value of the limit using a graph, consider the function \( f(x) = \left(1 + \frac{2}{x}\right)^x \). As \( x \to \infty \), observe how the function behaves. Notice that it approaches a certain value, reminiscent of the definition of Euler's number \( e^k \), where \( k \) is a constant. In this case, the graph would suggest that the function approaches \( e^2 \) (approximately 7.389).

Step 2: Apply l'Hospital's Rule Setup

Recognize the expression \( \lim _{x \rightarrow \, \infty}\left(1+\frac{2}{x}\right)^{x} \) as an indeterminate form of type \( 1^\infty \). Rewrite the function using the natural exponential function: \[ \lim_{x \to \infty} \left( 1 + \frac{2}{x} \right)^x = \exp \left( \lim_{x \to \infty} x \log \left( 1 + \frac{2}{x} \right) \right). \]

Step 3: Simplify the Limit Expression

Focus on computing the inner limit: \( \lim_{x \to \infty} x \log \left( 1 + \frac{2}{x} \right) \). Use the Taylor expansion \( \log(1 + a) \approx a - \frac{a^2}{2} \) for small \( a \), in this case \( \frac{2}{x} \). Therefore, \[ x \log \left( 1 + \frac{2}{x} \right) \approx x \left( \frac{2}{x} - \frac{\left( \frac{2}{x} \right)^2}{2} \right) = 2 - \frac{2}{2x}. \] As \( x \to \infty \), the expression simplifies to 2.

Step 4: Conclusion using l'Hospital's Rule

Since the limit of the expression inside the exponential becomes 2, use the continuous nature of the exponential function to evaluate: \[ \exp \left( \lim_{x \to \infty} \, x \log \left( 1 + \frac{2}{x} \right) \right) = e^2. \]

Key Concepts

Limit of a Function

The concept of the limit of a function is fundamental in calculus. It helps us understand the behavior of functions as they approach a certain input value. The limit tells us what value the function is approaching. For example, when we say \( \lim _{x \rightarrow \infty} f(x) \), we are interested in what happens to the function \( f(x) \) as \( x \) becomes arbitrarily large.
This is especially useful in finding the limiting behavior of functions where direct computation is complex or unwieldy.

• Limits help analyze functions at points of continuity and discontinuity.
• They provide insights into function behavior for values that are not directly computable.

Understanding limits is crucial in calculus both for academic success and practical application in sciences and engineering.

Indeterminate Forms

When we encounter expressions like \( \frac{0}{0} \), \( 1^\infty \), or \( \frac{\infty}{\infty} \), these are known as indeterminate forms. They don't provide a clear indication of what the limit might be.
Indeterminate forms suggest that more work is needed to evaluate the limit. In this problem, \( \lim _{x \rightarrow \infty}(1+\frac{2}{x})^{x} \) is an indeterminate form of \( 1^\infty \), which means that directly plugging in the values won't give the correct limit.

• Common indeterminate forms include \( 0/0 \) and \( \infty/\infty \).
• These forms require special techniques like l'Hospital's Rule or algebraic manipulation to resolve.

L'Hospital's Rule is a powerful tool used to compute limits of indeterminate forms by transforming them into forms that can be more easily analyzed.

Natural Exponential Function

The natural exponential function \( e \) is a mathematical constant approximately equal to 2.71828. It arises naturally in the study of continuous growth processes, such as compound interest and population growth. In the expression \( \exp \), we transform potentially unwieldy expressions into a form involving exponents.
Using the natural exponential function, we can rewrite complex exponential limits in a form that is easier to handle mathematically. For instance, rewriting \( \left(1 + \frac{2}{x}\right)^x \) using the natural exponential function, we get \( \exp \left(x \log \left(1 + \frac{2}{x}\right)\right) \), which simplifies our work in finding limits.

• \( e^x \) is the solution to the differential equation \( y' = y \).
• It is the base of natural logarithms, meaning \( \ln(e) = 1 \).

The natural exponential function is foundational in calculus because it simplifies handling exponential forms.

Taylor Series Expansion

The Taylor series expansion is a method of approximating functions using an infinite sum of terms based on the derivatives of the function at a single point. It is incredibly useful for approximating complex functions in a simpler form.
For example, the Taylor series for \( \log(1 + a) \) around 0 is \( a - \frac{a^2}{2} + \frac{a^3}{3} - \ldots \). This formula allows us to approximate logarithmic functions when the input is close to zero.
In our problem, when \( a = \frac{2}{x} \), the series simplifies the expression \( \log \left(1 + \frac{2}{x}\right) \), making it easier to find the limit.

• It provides a power series that approximates a function near a specific point.
• Useful in simplifying complex calculations involving limits or derivatives.

Using Taylor expansions is vital for working with otherwise unwieldy functions, making them more approachable for analysis.