Menu Close

Calculer-lim-x-2-x-2-x-2-6-x-2-

Tinku Tara 0 Comments
Pin




Question Number 216316 by a.lgnaoui last updated on 03/Feb/25
Calculer lim_(x→−2) ((x^2 +x−2)/( (√(6+x)) −2))
$$\mathrm{Calculer} \\ $$$$\mathrm{lim}_{\boldsymbol{\mathrm{x}}\rightarrow−\mathrm{2}} \frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}−\mathrm{2}}{\:\sqrt{\mathrm{6}+\boldsymbol{\mathrm{x}}}\:−\mathrm{2}} \\ $$
Answered by A5T last updated on 03/Feb/25
((x^2 +x−2)/( (√(6+x))−2))=(((x+2)(x−1)((√(6+x))+2))/(2+x)) =(x−1)((√(6+x))+2) ⇒lim_(x→−2) (((x^2 +x−2)/( (√(6+x))−2)))=−3×4=−12
$$\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{2}}{\:\sqrt{\mathrm{6}+\mathrm{x}}−\mathrm{2}}=\frac{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}−\mathrm{1}\right)\left(\sqrt{\mathrm{6}+\mathrm{x}}+\mathrm{2}\right)}{\mathrm{2}+\mathrm{x}} \\ $$$$=\left(\mathrm{x}−\mathrm{1}\right)\left(\sqrt{\mathrm{6}+\mathrm{x}}+\mathrm{2}\right) \\ $$$$\Rightarrow\underset{\mathrm{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\left(\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{2}}{\:\sqrt{\mathrm{6}+\mathrm{x}}−\mathrm{2}}\right)=−\mathrm{3}×\mathrm{4}=−\mathrm{12} \\ $$
Answered by MrGaster last updated on 04/Feb/25
lim_(x→2) (((x−1)(x+2))/( (√(6+x))−2)) =lim_(x→−2) (((x−1)(x+2)((√(6+x))+2))/(((√(6+x))−2)((√(6+x))+2))) =lim_(x→−2) (((x−1)(x−2)((√(6+x))−2))/(6+x−4)) =lim_(x→−2) (((x−1)(x−2)((√(6−x))+2))/(x−2)) =lim_(x→−2) (x−1)((√(6+x))+2) =(−2−1)((√(6−2))+2) =(−3)((√4)+2) =(−3)(2+2) =(−3)(4) =−12
$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{\:\sqrt{\mathrm{6}+{x}}−\mathrm{2}} \\ $$$$=\underset{{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\frac{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left(\sqrt{\mathrm{6}+{x}}+\mathrm{2}\right)}{\left(\sqrt{\mathrm{6}+{x}}−\mathrm{2}\right)\left(\sqrt{\mathrm{6}+{x}}+\mathrm{2}\right)} \\ $$$$=\underset{{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\frac{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left(\sqrt{\mathrm{6}+{x}}−\mathrm{2}\right)}{\mathrm{6}+{x}−\mathrm{4}} \\ $$$$=\underset{{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\frac{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left(\sqrt{\mathrm{6}−{x}}+\mathrm{2}\right)}{{x}−\mathrm{2}} \\ $$$$=\underset{{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\left({x}−\mathrm{1}\right)\left(\sqrt{\mathrm{6}+{x}}+\mathrm{2}\right) \\ $$$$=\left(−\mathrm{2}−\mathrm{1}\right)\left(\sqrt{\mathrm{6}−\mathrm{2}}+\mathrm{2}\right) \\ $$$$=\left(−\mathrm{3}\right)\left(\sqrt{\mathrm{4}}+\mathrm{2}\right) \\ $$$$=\left(−\mathrm{3}\right)\left(\mathrm{2}+\mathrm{2}\right) \\ $$$$=\left(−\mathrm{3}\right)\left(\mathrm{4}\right) \\ $$$$=−\mathrm{12} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *