Problem 61 - Calculus for Biology and Medicine

Question

Let \(f(x)=\ln x\). We know that \(f^{\prime}(x)=\frac{1}{x}\). We will use th fact and the definition of derivatives to show that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$ (a) Use the definition of the derivative to show that $$ f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h} $$ (b) Show that (a) implies that $$ \ln \left[\lim _{h \rightarrow 0}(1+h)^{1 / h}\right]=1 $$ (c) Set \(h=\frac{1}{n}\) in (b) and let \(n \rightarrow \infty\). Show that this implies that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$

Step-by-Step Solution

Answer

\( \lim_{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e \) using the definition of a derivative at 1.

Step 1: Establish the derivative's definition

According to the definition of a derivative, \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]We apply this definition to the function \(f(x) = \ln x\):\[ f'(1) = \lim_{h \to 0} \frac{\ln(1+h) - \ln 1}{h} = \lim_{h \to 0} \frac{\ln(1+h)}{h} \]Since \(\ln 1 = 0\), this simplifies the second part of the derivative definition.

Step 2: Using the derivative value

We know \(f'(x) = \frac{1}{x}\), and specifically, \(f'(1) = 1\). Applying this to the expression, we have:\[ f'(1) = \lim_{h \to 0} \frac{\ln(1+h)}{h} = 1 \]

Step 3: Transform the expression

We manipulate the expression from Step 2 to develop another form:\[ \ln \left[ \lim_{h \to 0} (1+h)^{1/h} \right] = \lim_{h \to 0} \frac{\ln(1+h)}{h} = 1 \]This equation implies:\[ \ln(A) = 1 \]where \( A = \lim_{h \to 0} (1+h)^{1/h} \). Since the natural log of \(A\) is 1, \(A = e\).

Step 4: Substitution and conclusion

Set \( h = \frac{1}{n} \). As \( n \to \infty \), \( h \to 0 \). Substitute \( h \) into the expression from Step 3:\[ \lim_{h \to 0} (1+h)^{1/h} = \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^{n} \]Since \( \ln \left[ \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^{n} \right] = 1 \), we determine that:\[ \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^{n} = e \]

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. Think of a derivative as a tool to find the "rate of change" or the "slope" of a function at a given point.
In mathematical terms, if you have a function \( f(x) \), its derivative at a point \( x \) is given by the limit:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This formula helps to understand how the function behaves in very small changes around the point.
When applying it to the natural logarithm function \( f(x) = \ln(x) \), you'd find that its derivative is \( f'(x) = \frac{1}{x} \). This shows how the slope of \( \ln(x) \) becomes smaller as \( x \) increases. Specifically, for \( x = 1 \), this derivative equals 1, representing the rate at which \( \ln(x) \) changes precisely at \( x = 1 \).
Understanding this is crucial in proving the limits associated with exponential growth, which brings us to explore the next concept, natural logarithms.

Delving into Natural Logarithms

Natural logarithms, often denoted \( \ln(x) \), are the logarithms to the base \( e \), where \( e \approx 2.71828 \). They're widely used in mathematics, especially in calculus and growth-related problems.
A natural logarithm tells us how much exponentiation we need to get a particular number from \( e \). For instance, \( \ln(e) = 1 \) because \( e^1 = e \).
The power of natural logarithms lies in their ability to simplify multiplicative relationships into additive ones, making them convenient for solving equations involving exponential functions.
In our exercise, we use the property \( \ln \left[ \lim_{h \to 0}(1+h)^{1/h} \right] = 1 \) to demonstrate that exponentiating yields \( e \). This effectively means that the base of the natural log itself emerges when the limit process is applied to exponential growth setups.
Natural logarithms are therefore crucial for understanding the continuous compound growth, as they often emerge in expressions that define such limits.

Exploring Exponential Functions

Exponential functions, written as \( f(x) = a^x \), are functions where a constant base \( a \) is raised to a variable power \( x \). These functions describe processes that rapidly increase or decrease, depending on the base value and the context.
The special case of \( e^x \) is particularly important due to \( e \)'s unique properties.
The exponential function \( f(x) = e^x \) is renowned for being its own derivative, meaning \( \frac{d}{dx} e^x = e^x \). This trait allows for elegant solutions in differential equations and natural growth processes.
In our context, the expression \( \left(1 + \frac{1}{n}\right)^n \) approximates \( e \) as \( n \to \infty \). This ties into Euler's number \( e \) emerging naturally in various mathematical contexts, notably in compound interest calculations and population growth models.
When we talk about limits involving exponential forms, like \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \), it's an illustration of how continuous compounding comes into play, highlighting the significance of \( e \) in continuous growth phenomena.
Exponential functions, therefore, provide the backbone for modeling numerous real-world processes, underscoring the importance of understanding these functions for solving practical problems.