Question

$\lim_{x \to \ 3} \frac{f(x)-f(3)}{x-3} sendo f(x)=5x^2$

Answer 1

$f\left(x \right )=5x^{2}$

$\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{f\left(x \right )-f\left(3 \right )}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{5x^{2}-5\cdot \left(3 \right )^{2}}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{5x^{2}-5\cdot 9}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{5x^{2}-45}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{5\cdot \left(x^{2}-9 \right )}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{5\cdot \left(x+3 \right )\cdot \left(x-3 \right )}{x-3}\\ \\ =\underset{x \to 3}{\mathrm{\ell im}}\;5\cdot \left(x+3 \right )\\ \\ =5\cdot \left(3+3 \right )\\ \\ =5\cdot 6\\ \\ =30\\ \\ \\ \boxed{\underset{x \to 3}{\mathrm{\ell im}}\;\dfrac{f\left(x \right )-f\left(3 \right )}{x-3}=30}$