Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 180106 by cherokeesay last updated on 07/Nov/22

lim_(x→∞) (√(x^2 +x))−x

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\sqrt{{x}^{\mathrm{2}} +{x}}−{x} \\ $$

Answered by a.lgnaoui last updated on 07/Nov/22

(√(x^2 +x)) −x=((((√(x^2 +x)) −x)((√(x^2 +x)) +x))/( (√(x^2 +x)) +x))  (x∈(]−∞,0]∪[−1,+∞[)  =(x/( (√(x^2 +x)) +x)) =(x/(x((√(1+(1/x))) +1)))  =(1/( (√(1+(1/x))) +1))  lim_(x→∞) (√(x^2 +x)) −x=lim_(x→∞) (1/( (√(1+(1/x))) +1))=(1/2)

$$\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:−\mathrm{x}=\frac{\left(\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:−\mathrm{x}\right)\left(\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:+\mathrm{x}\right)}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:+\mathrm{x}}\:\:\left(\mathrm{x}\in\left(\right]−\infty,\mathrm{0}\right]\cup\left[−\mathrm{1},+\infty\left[\right)\right. \\ $$$$=\frac{\mathrm{x}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:+\mathrm{x}}\:=\frac{\mathrm{x}}{\mathrm{x}\left(\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}}\:+\mathrm{1}\right)}\:\:=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}}\:+\mathrm{1}} \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow\infty} \sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\:−\mathrm{x}=\mathrm{lim}_{\mathrm{x}\rightarrow\infty} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}}\:+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by cherokeesay last updated on 07/Nov/22

very nice,  thank you sir.

$${very}\:{nice}, \\ $$$${thank}\:{you}\:{sir}. \\ $$

Answered by CElcedricjunior last updated on 07/Nov/22

(1/2)

$$\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Answered by mr W last updated on 07/Nov/22

=lim_(x→∞) (1/(1/x))((√(1+(1/x)))−1)  =lim_(t→0) (1/t)((√(1+t))−1)  =lim_(t→0) (1/t)(1+(t/2)+o(t^2 )−1)  =lim_(t→0) ((1/2)+o(t))  =(1/2)

$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\frac{\mathrm{1}}{{x}}}\left(\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{x}}}−\mathrm{1}\right) \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{t}}\left(\sqrt{\mathrm{1}+{t}}−\mathrm{1}\right) \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{t}}\left(\mathrm{1}+\frac{{t}}{\mathrm{2}}+{o}\left({t}^{\mathrm{2}} \right)−\mathrm{1}\right) \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{2}}+{o}\left({t}\right)\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by cherokeesay last updated on 07/Nov/22

Nice, thank you master.

$${Nice},\:{thank}\:{you}\:{master}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com