$\lim\limits_{h\to 0} \dfrac{\ln(3+h)-\ln (3)}{h}$ is

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The definition of a derivative is give as the limit of the difference quotient:

$$\lim \limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

From our question, we can derive the function and the specified $x$ coordinate.

$$\lim\limits_{h\to 0} \dfrac{\ln(3+h)-\ln (3)}{h}$$ $$f(x)=\ln x \hskip{2em} x=3$$

Since we are looking for the derivative at $x=3$,

$$f'(x)=\frac{1}{x}$$ $$f'(3)=\boxed{\frac{1}{3}}$$