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: $\lim_{{x \to 0}} \frac{x \cdot 5^x}{x \ln(5)} = \lim_{{x \to 0}} \frac{x}{x} \cdot \frac{5^x}{\ln(5)}$