Problem 85 - Thomas Calculus

Question

Show that $$\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r}$$

Step-by-Step Solution

Answer

The limit is \( e^r \).

Step 1: Understand the Limit Definition

Recognize that the expression \( \lim_{k\rightarrow \infty} \left(1+\frac{r}{k}\right)^{k} \) is a well-known limit that defines the exponential function. Specifically, as \( k \to \infty \), the expression \( \left(1+\frac{r}{k}\right)^k \) approaches \( e^r \).

Step 2: Express in Exponential Form

Rewrite the expression using the natural exponential function. Recognize that the expression \( \left(1+\frac{r}{k}\right)^{k} \) can be rewritten as \( \exp\left(k \ln\left(1+\frac{r}{k}\right)\right) \).

Step 3: Use Logarithmic Approximation

For values where \( x \) is close to 0, use the approximation \( \ln(1+x) \approx x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots \). Applying this to \( \ln\left(1+\frac{r}{k}\right) \), we have \( \ln\left(1+\frac{r}{k}\right) \approx \frac{r}{k} - \frac{1}{2}\left(\frac{r}{k}\right)^2 + \ldots \).

Step 4: Simplify the Expression

Multiply the approximation by \( k \):\( k \ln\left(1+\frac{r}{k}\right) \approx k \left(\frac{r}{k} - \frac{1}{2}\left(\frac{r}{k}\right)^2 + \ldots \right) = r - \frac{r^2}{2k} + \ldots \).

Step 5: Take the Limit as \( k \to \infty \)

Observe that as \( k \to \infty \), the terms \(-\frac{r^2}{2k} + \ldots \) approach zero, which simplifies the expression to \( r \). Thus, the limit becomes:\( \lim_{k\rightarrow \infty} \exp\left(r - \frac{r^2}{2k} + \ldots\right) = \exp(r) \).

Step 6: Conclude That the Limit is \( e^r \)

Since \( \exp(r) = e^r \), we conclude that:\( \lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r} \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable appears in the exponent. One of the most important forms is the natural exponential function expressed as \( e^x \), where \( e \) is the mathematical constant approximately equal to 2.71828. Exponential functions have unique properties, especially their rate of growth, which is determined by the base. In this case, the base \( e \) results in a function that grows proportionally at every point, making it a key concept in calculus and natural phenomena.

• Exponential functions are essential in modeling population growth, radioactive decay, and many other natural processes.
• The limit expression given in the exercise, \( \left(1+\frac{r}{k}\right)^{k} \), is a classic example of approaching an exponential function as \( k \) becomes very large.

The goal here is to understand how limits can define the exponential function specifically through repeated compounding, which is a foundation of continuous growth models.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the inverse of the natural exponential function \( e^x \). Understanding the natural logarithm helps us deal with the exponential function and solve equations involving exponentials. It is based on Euler's number \( e \) and is hugely significant in calculus due to its ability to simplify the differentiation and integration of exponential functions.

• The natural logarithm is used to reverse the effect of an exponential, making expressions like \( \ln(e^x) \) simply equal to \( x \).
• In the exercise, \( \ln(1+\frac{r}{k}) \) is approximated to pave the way for expressing the original limit in terms of exponentials.

Grasping how the natural logarithm can simplify complex growth problems is key in learning how exponential growth can be represented in a more approachable form.

Limit Definition

In mathematics, a limit is a value that a function or sequence "approaches" as the input or index approaches some value. The limit is a fundamental notion in calculus and provides a way to understand the behavior of functions as they approach specific points. In this problem, the limit \( \lim_{k \rightarrow \infty} \left(1+\frac{r}{k}\right)^{k} = e^{r} \) is a key aspect of defining exponential growth through continuous compounding.

• Understanding limits helps in grasping how functions behave towards specific inputs, which is crucial in calculus.
• This problem showcases a special limit that underpins the natural exponential function \( e^x \).

Recognizing and calculating limits allows us to appreciate the behavior of functions not just at specific points, but as they trend towards infinity, enabling deeper insights into continuous growth models.

Logarithmic Approximation

Logarithmic approximation is a technique used to simplify complex logarithmic expressions, making them more manageable, especially as certain values approach zero. The formula \( \ln(1+x) \approx x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \) is often used when \( x \) is small. This approximation is vital when expressing limits involving exponential functions.

• This method simplifies the understanding and calculation of limits, especially when the initial expressions are complex.
• In the exercise, it allows us to express \( \ln(1+\frac{r}{k}) \) in a form that can be easily managed as \( k \) approaches infinity.

Using logarithmic approximation transforms challenging expressions into simpler sequences, enabling us to zero in on the behavior of functions as they reach limits.