Question

Va rog !!!
f: D -> R ; f(x)= ln(2x+3)
Calculati $\lim_{x \to \33} \frac{f(x) - f(3)}{x-3}$

Answer 1

$f(x) = \ln(2x+3)$

$f'(x) = \dfrac{(2x+3)'}{2x+3} = \dfrac{2}{2x+3}$

$\lim\limits_{x\to 3}\dfrac{f(x)-f(3)}{x-3}\overset{\frac{0}{0}(L'H.)}{=} \lim\limits_{x\to 3}\dfrac{\left[f(x)-f(3)\right]'}{(x-3)'}=$

$= \lim\limits_{x\to 3}\dfrac{f'(x) - 0}{1-0} = \lim\limits_{x\to 3}f'(x) = f'(3) = \dfrac{2}{2\cdot 3+3} =$

$= \boxed{\dfrac{2}{9}}$