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The given problem is:

\[
\lim_{{x \to 0}} \frac{x \cdot 3^x}{3^x - 1}
\]

To apply L'Hôpital's Rule, we first need to check if the limit results in an indeterminate form like \( \frac{0}{0} \). Let’s check that:

1. As \( x \to 0 \), \( 3^x \to 1 \).
2. The numerator \( x \cdot 3^x \to 0 \).
3. The denominator \( 3^x - 1 \to 0 \).

Since we have the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which involves taking the derivative of the numerator and denominator.

### Step 1: Derivatives
- Numerator: \( f(x) = x \cdot 3^x \)
  \[
  f'(x) = \frac{d}{dx}(x \cdot 3^x) = 3^x + x \cdot 3^x \ln(3)
  \]
- Denominator: \( g(x) = 3^x - 1 \)
  \[
  g'(x) = 3^x \ln(3)
  \]

### Step 2: Apply L'Hôpital's Rule
Now, we calculate the limit of the ratio of the derivatives:

\[
\lim_{{x \to 0}} \frac{3^x + x \cdot 3^x \ln(3)}{3^x \ln(3)}
\]

At \( x = 0 \):
- \( 3^x = 1 \),
- The numerator becomes \( 1 + 0 = 1 \),
- The denominator becomes \( \ln(3) \).

Thus, the limit becomes:

\[
\frac{1}{\ln(3)}
\]

### Final Answer:
\[
\lim_{{x \to 0}} \frac{x \cdot 3^x}{3^x - 1} = \frac{1}{\ln(3)}
\]

Would you like a detailed breakdown of any step, or do you have any additional questions?

### 5 Related Questions:
1. How does L'Hôpital's Rule work for other indeterminate forms, like \( \infty / \infty \)?
2. Can we solve this limit using a series expansion instead of L'Hôpital's Rule?
3. How would the limit change if the base of the exponential function was different, like \( e^x \)?
4. What is the derivative of the function \( a^x \) in general, where \( a \) is any positive constant?
5. Can you explain the conditions under which L'Hôpital's Rule is applicable?

### Tip:
When applying L'Hôpital's Rule, make sure to check the indeterminate form before taking derivatives. Only certain forms like \( 0/0 \) and \( \infty/\infty \) allow the use of this rule.

Find the limit: \( \lim_{{x \to 0}} \frac{x \cdot 3^x}{3^x - 1} \) using L'Hôpital's Rule.

Solve the limit problem \( \lim_{{x \to 0}} \frac{x \cdot 3^x}{3^x - 1} \) using L'Hôpital's Rule. This step-by-step solution applies calculus concepts such as the derivative of exponential functions and the rule for indeterminate forms. Ideal for university students studying Calculus I.

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