$\lim _\limits{x \rightarrow 0} \dfrac{a^x-b^x}{x} \text { is equal to }$

The given limit is $\lim_{x \rightarrow 0} \dfrac{a^x - b^x}{x}$.

Rewrite the expression by adding and subtracting $1$ in the numerator:

$\lim_{x \rightarrow 0} \dfrac{(a^x - 1) - (b^x - 1)}{x}$

Using the property of limits, this can be split into two separate limits:

$\lim_{x \rightarrow 0} \dfrac{a^x - 1}{x} - \lim_{x \rightarrow 0} \dfrac{b^x - 1}{x}$

Recall the standard limit formula $\lim_{x \rightarrow 0} \dfrac{k^x - 1}{x} = \ln k$.

Applying this formula to both terms:

$\ln a - \ln b$

Using the logarithmic property $\ln a - \ln b = \ln \left( \dfrac{a}{b} \right)$, the result is $\log \left( \dfrac{a}{b} \right)$.

Answer: $\log \dfrac{a}{b}$