Question

Cacular o limite de: 

 

lim x->5      ( √x - √5 ) / ( √x+5 - √10 ) 

 

Segundo o livro do Guidorizzi o resultado é √2.

Gostaria de saber o desenvolvimento desse calculo.

 

WolframAlpha (lim ((sqrt x) - (sqrt 5)) / ((sqrt x+5) - (sqrt 10)), x=5)

Answer 1

$\frac{ \sqrt{x}- \sqrt{5}}{\sqrt{x+5}- \sqrt{10}}=\frac{ \sqrt{x}- \sqrt{5}}{\sqrt{x+5}- \sqrt{10}}.\frac{\sqrt{x+5}+\sqrt{10}}{\sqrt{x+5}+\sqrt{10}}=\frac{ (\sqrt{x}- \sqrt{5})(\sqrt{x+5}+\sqrt{10})}{\sqrt{x+5}^2- \sqrt{10}^2}$

$=\frac{ (\sqrt{x}- \sqrt{5})(\sqrt{x+5}+\sqrt{10})}{x+5-10}=\frac{ (\sqrt{x}- \sqrt{5})(\sqrt{x+5}+\sqrt{10})}{x-5}.\frac{(\sqrt{x}+\sqrt{5})}{(\sqrt{x}+\sqrt{5})}$

$=\frac{ (\sqrt{x}^2- \sqrt{5}^2)(\sqrt{x+5}+\sqrt{10})}{(x-5)(\sqrt{x}+\sqrt{5})}=\frac{(x-5)(\sqrt{x+5}+\sqrt{10})}{(x-5)(\sqrt{x}+\sqrt{5})}=\frac{\sqrt{x+5}+\sqrt{10}}{\sqrt{x}+\sqrt{5}}$

$\lim_{n \to 5} \frac{ \sqrt{x}- \sqrt{5}}{\sqrt{x+5}- \sqrt{5}}=\lim_{n \to 5} \frac{\sqrt{x+5}+\sqrt{10}}{\sqrt{x}+\sqrt{5}}=\frac{\sqrt{5+5}+\sqrt{10}}{\sqrt{5}+\sqrt{5}}$

$=\frac{\sqrt{10}+\sqrt{10}}{\sqrt{5}+\sqrt{5}}=\frac{2\sqrt{10}}{2\sqrt{5}}=\sqrt{\frac{10}{5}}=\sqrt{2}$


Guidorizzi, quanto tempo!!!! (Época boa)

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