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Question Number 120843 by bramlexs22 last updated on 03/Nov/20

 lim_(x→3)  ((√(x+9−6(√x)))/( (√x)−3)) ?

$$\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}+\mathrm{9}−\mathrm{6}\sqrt{\mathrm{x}}}}{\:\sqrt{\mathrm{x}}−\mathrm{3}}\:? \\ $$

Commented by Dwaipayan Shikari last updated on 03/Nov/20

((√(12−6(√3)))/( (√3)−3))=−1

$$\frac{\sqrt{\mathrm{12}−\mathrm{6}\sqrt{\mathrm{3}}}}{\:\sqrt{\mathrm{3}}−\mathrm{3}}=−\mathrm{1} \\ $$

Answered by Jamshidbek2311 last updated on 03/Nov/20

((√(3+9−6(√3)))/((√3)−3))=((√(3(4−2(√3))))/((√3)−3))=((√(3((√3)−1)^2 ))/((√3)−3))=  ((((√3)−1)(√3))/((√3)−3))=((3−(√3))/((√3)−3))=−1

$$\frac{\sqrt{\mathrm{3}+\mathrm{9}−\mathrm{6}\sqrt{\mathrm{3}}}}{\sqrt{\mathrm{3}}−\mathrm{3}}=\frac{\sqrt{\mathrm{3}\left(\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}\right)}}{\sqrt{\mathrm{3}}−\mathrm{3}}=\frac{\sqrt{\mathrm{3}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)^{\mathrm{2}} }}{\sqrt{\mathrm{3}}−\mathrm{3}}= \\ $$$$\frac{\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)\sqrt{\mathrm{3}}}{\sqrt{\mathrm{3}}−\mathrm{3}}=\frac{\mathrm{3}−\sqrt{\mathrm{3}}}{\sqrt{\mathrm{3}}−\mathrm{3}}=−\mathrm{1} \\ $$

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