Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \lim _{h \rightarrow 0} \frac{e^{1+h}-e}{h} $$

Step-by-Step Solution

Answer

The limit as \(h\) approaches 0 is \(e\).

Step 1: Apply L'Hopital’s Rule

Begin by applying L'Hopital’s Rule, which states that the limit of a function as it approaches a particular value can be found by taking the derivative of the numerator and denominator separately. So differentiate \(e^{1+h}\) with respect to \(h\) to get \(e^{1+h}\), and differentiate \(h\) with respect to \(h\) to get 1.

Step 2: Simplify the Function

After differentiating, the function simplifies to: \(\frac{e^{1+h}}{1}\). Now it's easy to find the limit by directly substituting \(h = 0\) to the resulting function.

Step 3: Substitute \(h=0\)

Replace \(h\) with 0 in \(\frac{e^{1+h}}{1}\) to get \(e^1\), or simply \(e\).

Key Concepts

Understanding Limits of Functions

The concept of limits is foundational in calculus and helps us understand the behavior of functions as they approach a specific point. Imagine you're walking towards a tree. The closer you get, the clearer you see it. Similarly, finding the limit of a function as it approaches a certain value involves understanding what value the function approaches as the input gets increasingly close to that point.

In our exercise, we're tasked with finding the limit as h approaches 0 for the function \(\frac{e^{1+h}-e}{h}\). It's like asking: As h gets really small, what value does our function get close to? However, since the presence of h in the denominator can lead to division by zero, we need a method to handle this indeterminate form, and that's where L'Hopital's Rule becomes a powerful tool.

The Role of Derivatives in Calculus

A derivative represents the rate of change of a function with respect to a variable. It's like measuring how fast you're walking at any given moment. If we take our walking analogy a step further, the derivative tells us our pace - are we slowing down, speeding up, or moving at a constant speed?

In the context of L'Hopital's Rule, we use derivatives to deal with limits that result in an indeterminate form like 0/0. By taking the derivative of the numerator and the denominator separately, we often simplify the function and eliminate the indeterminate form, making it possible to find the limit. In our textbook example, by differentiating the numerator \(e^{1+h}\) and the denominator \(h\), we clarified the limit's value, which would not have been directly observable otherwise.

Exponential Functions and Their Behavior

An exponential function is a powerful mathematical concept where a constant base is raised to a variable exponent. It describes a situation where a quantity grows or decays at a rate proportional to its current value—think of how interests compound in a bank account, or how populations grow.

In our exercise, the function \(e^{1+h}\) is exponential, with e (approximately 2.71828) being the base and 1+h the exponent. As we investigate the limit for h approaching 0, we're essentially looking at the behavior of this exponential function as the exponent gets very close to 1. Upon applying L'Hopital's Rule and taking the derivative, we see that the explosive growth inherent in exponential functions doesn't complicate finding this limit, simplifying to just \(e\) as h goes to zero.