Understanding the Limit of (eh - 1) / h as h Approaches 0

Introduction

This article explores the limit of the function $frac{e^h - 1}{h}$ as $h$ approaches 0. We will discuss three different approaches to solve this limit: the Taylor series expansion, the definition of the derivative, and L'H?pital's Rule. This topic is crucial for understanding the derivative of the exponential function at 0, which is a fundamental concept in calculus.

1. Taylor Series Expansion

The first method involves using the Taylor series expansion of the exponential function.

Step 1: Expand $e^h$ using the Taylor series around 0. [e^h 1 h frac{h^2}{2!} frac{h^3}{3!} cdots]

Step 2: Substitute the series into the function and simplify.

[frac{e^h - 1}{h} frac{1 h frac{h^2}{2!} frac{h^3}{3!} cdots - 1}{h} frac{h frac{h^2}{2!} frac{h^3}{3!} cdots}{h} 1 frac{h}{2!} frac{h^2}{3!} cdots]

Step 3: Take the limit as $h$ approaches 0.

[lim_{h to 0} left(1 frac{h}{2!} frac{h^2}{3!} cdots right) 1]

Conclusion: The limit of $frac{e^h - 1}{h}$ as $h$ approaches 0 is 1.

2. Definition of the Derivative

The second method involves using the definition of the derivative of the exponential function.

Step 1: Use the definition of the derivative of $f(x) e^x$, where the derivative at any point $x$ is given by the limit

[f'(x) lim_{h to 0} frac{e^{x h} - e^x}{h}]

Step 2: Apply this definition at $x 0$ and recognize that $e^0 1$.

[lim_{h to 0} frac{e^{0 h} - e^0}{h} lim_{h to 0} frac{e^h - 1}{h}]

Conclusion: According to the definition, the limit of $frac{e^h - 1}{h}$ as $h$ approaches 0 is the derivative of the exponential function at 0, which is $1$.

3. L'H?pital's Rule

The third method involves using L'H?pital's Rule, which is applicable for limits of the form $frac{0}{0}$. This rule states that if the limit of the ratio of two functions results in an indeterminate form, then the limit of the ratio of their derivatives is the same, provided the limit of the derivatives exists.

Step 1: Identify the limit as an indeterminate form $frac{0}{0}$. [lim_{h to 0} frac{e^h - 1}{h} frac{0}{0}]

Step 2: Apply L'H?pital's Rule to differentiate the numerator and the denominator with respect to $h$.

[lim_{h to 0} frac{frac{d}{dh}(e^h - 1)}{frac{d}{dh}(h)} lim_{h to 0} frac{e^h}{1} lim_{h to 0} e^h e^0 1]

Conclusion: The limit of $frac{e^h - 1}{h}$ as $h$ approaches 0 is 1.

Verification by Substitution

To further verify, we can substitute small values of $h$ into the function to observe the behavior:

[begin{array}{c|c} text{h} frac{e^h - 1}{h} hline 0.1 0.994594 0.01 0.999499 0.001 0.999950 0.0001 0.999995 -0.01 0.999501 -0.001 0.999950 -0.0001 0.999995 end{array}]

As $h$ approaches 0, the value of $frac{e^h - 1}{h}$ approaches 1, which confirms our theoretical results.

Conclusion: The limit of the function $frac{e^h - 1}{h}$ as $h$ approaches 0 is 1. This can be shown using Taylor series expansion, the definition of the derivative, and L'H?pital's Rule. Each method provides a different perspective on why this is true, and together they reinforce the importance of understanding foundational concepts in calculus.