Problem 55 - Essential Calculus

Question

Think About It In Exercises 55-58, L'Hopital's Rule is used incorrectly. Describe the error. \(\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}}=\lim _{x \rightarrow 0} \frac{2 e^{2 x}}{e^{x}}=\lim _{x \rightarrow 0} 2 e^{x}=2\)

Step-by-Step Solution

Answer

The error in the student's solution is that they applied the L'Hopital's Rule on the original limit, but that was inappropriate as the original limit did not have an indeterminate form of 0/0 or ∞/∞. The correct approach would be to first rewrite the expression, especially the numerator, in order to properly evaluate the limit.

Step 1: Identify the original limit

Observing the given exercise, \( \lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}} \), it is not an indeterminate form of 0/0 or ∞/∞. So, it's clear that L'Hopital's Rule cannot be applied initially.

Step 2: Understand the mistake in the used approach

In the student's solution, they incorrectly applied L'Hopital's Rule on the original limit. They derived \( \lim _{x \rightarrow 0} \frac{2 e^{2 x}}{e^{x}} \) and then further simplified this as \( \lim _{x \rightarrow 0} 2 e^{x}=2 \). However, as analyzed in Step 1, L'Hopital's Rule is not appropriate for the original limit as it isn't indeterminate of the form 0/0 or ∞/∞.

Step 3: Correct Approach to solve the original limit

The correct approach is to apply elementary algebra and rewrite the original expression. For \( x \rightarrow 0 \), the denominator \( e^{x} \) is clearly tending to 1. So, it is the numerator that requires rewriting for limit evaluation. This could be a subsequent step in coming to the right solution

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change, which is essentially about movement and growth. Two foundational concepts within calculus are differentiation and integration, dealing with rates of change and accumulated quantities respectively. In the context of the given exercise, calculus facilitates the understanding of limits, which are fundamental in defining both derivatives and integrals.

One of the powerful tools in calculus for dealing with limits is L'Hopital's Rule, which is used to resolve indeterminate forms by converting them into a potentially easier form to evaluate. It's important to first verify that the indeterminate form is a valid candidate before applying this rule, as each tool within calculus has its appropriate application domain. We'll explore the correct use of L'Hopital's Rule in the subsequent sections.

Limits

The concept of limits is at the core of calculus and needs to be understood in order to proceed to more advanced topics. A limit describes the value that a function approaches as the input approaches some value. Limits can tell us about the behavior of functions at points where they're not explicitly defined or where they exhibit interesting behavior.

In the textbook exercise, we're looking at the limit as the variable x approaches zero. Understanding limits also means recognizing when they are in an indeterminate form and knowing that extra manipulation or different methods, such as L'Hopital's Rule, may be required to find the limit's value.

Indeterminate Forms

Indeterminate forms occur when the limit of a function cannot be directly determined and may seem to have different potential values. Common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0\times\infty\), \(\infty-\infty\), \(1^\infty\), \(0^0\), and \(\infty^0\).

The expression provided in the exercise, initially appears not to be an indeterminate form because when x approaches zero, the numerator approaches one and the denominator approaches one as well, giving a quick evaluation of 1. The student's assumption that L'Hopital's Rule applies was incorrect; this highlights the importance of accurately identifying indeterminate forms. In this case, a direct substitution gives the correct limit without the need for L'Hopital's Rule.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. They exhibit rapid growth or decay and are fundamental in modeling diverse phenomena such as population growth, radioactive decay, and interest compounding. These functions are of the form \(e^{tx}\), where e is the base of the natural logarithm.

In this exercise, we encounter exponential functions such as \(e^{2x}\) and \(e^x\). Understanding how these functions behave as x approaches a value is crucial for evaluating limits involving exponential expressions. It's pivotal to know that the exponential function gets asymptotically close to zero for negative exponents and grows without bound for positive exponents as x becomes very large. This characteristic behavior helps in determining the limits of exponential functions, especially when indeterminacies are not in play.