Problem 56 - Calculus and its applications

Question

Given that the derivative of \(f(x)=a^{x}\) is \(f^{\prime}(x)=a^{x}(\ln a),\) in Section 3.1 we showed that \(f^{\prime}(x)=a^{x} \cdot \lim _{h \rightarrow 0} \frac{a^{h}-1}{h}\). Thus, we can define \(\ln a=\lim _{h \rightarrow 0} \frac{a^{h}-1}{h}\). Use this definition to find each limit. $$ \lim _{h \rightarrow 0} \frac{e^{h}-1}{h} $$

Step-by-Step Solution

Answer

The limit \( \lim_{h \rightarrow 0} \frac{e^h - 1}{h} \) is 1.

Step 1: Understanding the Limit

We are asked to evaluate the limit \( \lim_{h \rightarrow 0} \frac{e^{h}-1}{h} \). Notice that this limit has a similar form to the definition provided: \( \ln a = \lim_{h \rightarrow 0} \frac{a^h - 1}{h} \). Here, \(a = e\), which is Euler's number, approximately equal to 2.71828.

Step 2: Substituting into the Definition

By substituting \(a = e\) into the provided definition \( \ln a = \lim_{h \rightarrow 0} \frac{a^h - 1}{h} \), we have \( \ln e = \lim_{h \rightarrow 0} \frac{e^h - 1}{h} \).

Step 3: Evaluating \(\ln e\)

We know that the natural logarithm of Euler's number, \( \ln e \), is equal to 1. Thus, we can substitute this value into our expression to get the limit. This means that \( \lim_{h \rightarrow 0} \frac{e^h - 1}{h} = 1 \).

Key Concepts

Limits

Limits are a fundamental concept in calculus and mathematical analysis. They help us understand the behavior of a function as it approaches a particular point or value. The notation \( \lim_{{x \to a}} f(x) \) represents the limit of \( f(x) \) as \( x \) approaches \( a \). Essentially, it's about getting as close as possible to a specific point without actually reaching it.

When dealing with limits, we often encounter situations that are indeterminate, such as division by zero. This is where the concept becomes crucial. By using limits, we can solve these indeterminate forms and find meaningful values.

In our exercise, we have the limit \( \lim_{{h \rightarrow 0}} \frac{e^h - 1}{h} \). This expression is similar to the definition of the derivative of exponential functions. Here, we explore how the function behaves as \( h \) approaches zero to find consistent behavior that helps us define the derivative precisely.

Natural Logarithms

Natural logarithms, denoted as \( \ln x \), are logarithms with Euler's number \( e \) as the base. They are vital in mathematics, particularly in calculus and complex analysis. Natural logarithms have distinct properties that simplify complex calculations, especially when dealing with growth processes such as exponential growth.

The relationship between derivatives and logarithms is evident in our exercise. The derivative of a function involving \( a^x \) brings in the natural logarithm as \( a^x(\ln a) \). This ties the behavior of exponential functions to their growth rate through differentiation.

• \( \ln e = 1 \), because \( e^1 = e \). This is a key feature, showing that the exponential function and its logarithm are inverses.
• If \( \ln a = \lim_{{h \rightarrow 0}} \frac{a^h - 1}{h} \), substituting \( e \) provides \( \ln e = \lim_{{h \rightarrow 0}} \frac{e^h - 1}{h} \), ensuring that the limit equals 1 as shown.

These properties help solve complex problems and understand exponential function growth.

Euler's Number

Euler’s number, denoted \( e \), is approximately 2.71828, and it's a fundamental constant in mathematics. It often appears when dealing with growth rates or processes that involve change. Euler's number uniquely defines the base of the natural logarithm and is crucial in the calculation of derivatives of exponential functions.

This number is often described as the "natural" base of logarithms because it arises naturally when calculating continuous growth. Its relationship with the infinite limit \( \lim_{{h \rightarrow 0}} (1 + \frac{1}{h})^h \) when \( h \to 0 \) exemplifies its role in growth and decay processes.

• \( e \) is unique because the function \( e^x \) is its own derivative, meaning \( \frac{d}{dx} e^x = e^x \). This property simplifies analysis hugely and cements \( e \)'s importance in calculus.
• In our exercise, the focus on \( \lim_{{h \rightarrow 0}} \frac{e^h - 1}{h} \) emphasizes \( e \)'s properties, as this limit helps define the derivative of \( e^x \) function.

Understanding Euler’s number enriches our knowledge of not only mathematics but also of natural phenomena described by exponential functions.