\( \lim\limits_{h\to 0} \dfrac{\ln(3+h)-\ln (3)}{h} \) 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 \(x=3\),

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