Problem 40 - Essential Calculus Early Transcendentals

Question

\(39-40=\) Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value. $$\lim _{x \rightarrow 0} \frac{5^{x}-4^{x}}{3^{x}-2^{x}}$$

Step-by-Step Solution

Answer

The limit is \(\frac{\ln\left(\frac{5}{4}\right)}{\ln\left(\frac{3}{2}\right)}\).

Step 1: Analyze the Indeterminate Form

First, substitute \(x = 0\) in the limit expression \( \frac{5^x - 4^x}{3^x - 2^x} \). This results in \( \frac{1 - 1}{1 - 1} = \frac{0}{0} \), indicating an indeterminate form. Therefore, we can apply l'Hospital's Rule.

Step 2: Apply l'Hospital's Rule

To resolve the indeterminate form, take the derivative of the numerator and the derivative of the denominator:The derivative of the numerator \(5^x - 4^x\) is \( 5^x \ln(5) - 4^x \ln(4) \).The derivative of the denominator \(3^x - 2^x\) is \( 3^x \ln(3) - 2^x \ln(2) \).

Step 3: Re-evaluate the Limit With Derivatives

Substitute back into the limit using these derivatives:\[ \lim_{x \to 0} \frac{5^x \ln(5) - 4^x \ln(4)}{3^x \ln(3) - 2^x \ln(2)} \]Plug \(x = 0\) into the differentiated expression:\[ \frac{1 \cdot \ln(5) - 1 \cdot \ln(4)}{1 \cdot \ln(3) - 1 \cdot \ln(2)} = \frac{\ln(5) - \ln(4)}{\ln(3) - \ln(2)} \]

Step 4: Simplify the Logarithmic Expression

Simplify the logarithmic expression \(\frac{\ln(5) - \ln(4)}{\ln(3) - \ln(2)}\):This simplifies further using properties of logarithms:\[ \frac{\ln\left(\frac{5}{4}\right)}{\ln\left(\frac{3}{2}\right)} \].

Step 5: Final Evaluation

The value of the limit is now expressed as \(\frac{\ln\left(\frac{5}{4}\right)}{\ln\left(\frac{3}{2}\right)}\), where both the numerator and denominator are constants.

Key Concepts

Understanding Indeterminate Forms

When you start solving limits in calculus, you might come across something known as an "indeterminate form." This specifically arises when substituting the limit value into an expression results in forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These results are 'indeterminate' because they don't provide a clear numerical answer.

In our exercise, by substituting \(x = 0\), we get \( \frac{0}{0} \), hence it's indeterminate. Recognizing these forms is crucial because it often means we need another method, like l'Hospital's Rule, to find the solution.

Indeterminate forms signal the need to dive deeper into the behavior of functions as they approach certain points. They tell us that direct evaluation isn't enough, and we must explore limits with more sophisticated techniques to find meaningful answers.

Using Derivatives in Calculus

Derivatives are a fundamental part of calculus. They provide information about the rate of change of functions. In the context of solving limits, like our exercise, derivatives help us break down expressions that are otherwise indeterminate.

When encountering an indeterminate form, l'Hospital's Rule allows us to use derivatives to strategically evaluate limits. This powerful rule states that if the limit \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can find the limit by taking the derivatives of the numerator and the denominator: \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \).

In our exercise, we took the derivative of both the top and bottom parts of the fraction, resulting in expressions involving logarithms, which simplifies the overall limit evaluation.

Simplifying with Logarithmic Expressions

Logarithmic expressions are often used in calculus to simplify complex derivatives, especially those involving exponential functions.

When we differentiate the expressions \(5^x\) and \(4^x\), we use logarithmic derivatives. The rule here is that the derivative of \(a^x\) is \(a^x \ln(a)\). Hence, the derivatives of the expressions become \(5^x \ln(5)\) and \(4^x \ln(4)\), respectively.

After applying l'Hospital's Rule, we introduce logarithms into our exercise, transforming the original problem into simpler terms that can be easily managed. The properties of logarithms, like \(\ln(a) - \ln(b) = \ln(\frac{a}{b})\), further simplify our results. Thus, we can express the limit in easily understandable terms: \(\frac{\ln\left(\frac{5}{4}\right)}{\ln\left(\frac{3}{2}\right)}\), making it clear and computable.