To approximate the limit

$\lim_{x \to 0} \frac{3^x - 2^x}{x},$

we will first analyze it using L'Hôpital's Rule, and then confirm the approximation using a graphical method.

Step 1: Analytic Approximation using L'Hôpital's Rule

Given the limit:

$\lim_{x \to 0} \frac{3^x - 2^x}{x},$

direct substitution $x = 0$ leads to the indeterminate form $\frac{0}{0}$. Therefore, we apply L'Hôpital's Rule, which requires us to differentiate the numerator and the denominator separately with respect to $x$.

The derivatives are:

• Numerator: The derivative of $3^x$ is $3^x \ln(3)$ and the derivative of $2^x$ is $2^x \ln(2)$.
• Denominator: The derivative of $x$ is 1.

Applying L'Hôpital's Rule gives:

$\lim_{x \to 0} \frac{3^x \ln(3) - 2^x \ln(2)}{1} = 3^0 \ln(3) - 2^0 \ln(2) = \ln(3) - \ln(2).$

Thus, the limit simplifies to:

$\ln(3) - \ln(2).$

Step 2: Numerical Value

Now, let's compute the numerical value:

$\ln(3) \approx 1.0986, \quad \ln(2) \approx 0.6931.$

So,

$\ln(3) - \ln(2) \approx 1.0986 - 0.6931 = 0.4055.$

Step 3: Graphical Confirmation

To confirm this approximation graphically, we can plot the function $f(x) = \frac{3^x - 2^x}{x}$ near $x = 0$ and observe the behavior of the function as $x$ approaches 0.

Given the results from the analytical approach and the numerical values, we expect the graph of $f(x)$ to approach approximately $0.4055$ as $x \to 0$.

Would you like further details on the graphical confirmation, or do you have any questions?

Related Questions to Expand Understanding:

1. How does L'Hôpital's Rule work, and when can it be applied?
2. What are some other examples of limits that lead to an indeterminate form?
3. How do logarithmic properties help simplify expressions like $\ln(3) - \ln(2)$?
4. Why is the derivative of $a^x$ equal to $a^x \ln(a)$?
5. What other methods can be used to approximate limits besides L'Hôpital's Rule?

Tip

When evaluating limits analytically, it’s helpful to recognize common indeterminate forms and have a set of tools, such as L'Hôpital's Rule, Taylor expansions, or series approximations, to approach the problem systematically.