Question

como calcular este limite?

lim┬( x→9) ( 9-x)/(√x-3)

Answer 1

Boa tarde Lauro!

Solução!

$\lim_{x \to 9} \dfrac{9-x}{ \sqrt{x} -3}$

Com o limite montado vamos fazer a racionalização do denominador com o sinal trocado.

$\lim_{x \to 9} \dfrac{9-x}{ \sqrt{x} -3} \times\dfrac{\sqrt{x} +3 }{ \sqrt{x} +3}\\\\\\ \lim_{x \to 9} \dfrac{(9-x)( \sqrt{x} +3)}{ \sqrt{x} . \sqrt{x} +3 \sqrt{x} -3 \sqrt{x} -9} \\\\\\\ \lim_{x \to 9} \dfrac{(9-x)( \sqrt{x} +3)}{ x -9}\\\\\\ Vamos ~~multiplicar~~ por~~ (-1)~~ o~~ numerador. \\\\\\\ \lim_{x \to 9} \dfrac{(9-x)( \sqrt{x} +3)(-1)}{ x -9}\\\\\\ \lim_{x \to 9} \dfrac{(-9+x)( -\sqrt{x} +3)}{ x -9}$

$\lim_{x \to 9} - (\sqrt{x}+3) \\\\\\ Substituindo~~o ~~valor~~x\\\\\\ \lim_{x \to 9} - (\sqrt{9}+3) \\\\\\ \lim_{x \to 9} - (3+3) \\\\\\ \lim_{x \to 9} - (6) \\\\\\ \lim_{x \to 9} -6$



$\boxed{Resposta:\lim_{x \to 9} \dfrac{9-x}{ \sqrt{x} -3}=-6}$

Boa tarde!
Bons estudos!