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To evaluate the limit:

\[
\lim_{{x \to 0}} \frac{x \cdot 5^x}{5^x - 1}
\]

we can approach it by analyzing the behavior of both the numerator and the denominator as \( x \to 0 \).

1. **Expanding \( 5^x \) near \( x = 0 \) using Taylor Series:**
   For \( x \) close to 0, we can approximate \( 5^x \) by its Taylor expansion:
   \[
   5^x \approx 1 + x \ln(5) + \frac{x^2 (\ln(5))^2}{2} + \dots
   \]
   So,
   \[
   5^x - 1 \approx x \ln(5)
   \]

2. **Rewrite the Limit:**
   Substitute this approximation in the limit:
   \[
   \lim_{{x \to 0}} \frac{x \cdot 5^x}{x \ln(5)} = \lim_{{x \to 0}} \frac{x}{x} \cdot \frac{5^x}{\ln(5)}
   \]
  

Evaluate the limit as x approaches 0 of (x * 5^x) / (5^x - 1).

Learn to evaluate the limit as x approaches 0 of (x * 5^x) / (5^x - 1) using Taylor series expansion and approximation techniques. This problem is suitable for advanced high school or college calculus students and covers key concepts such as exponential functions, Taylor series, and L'Hôpital's Rule.

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