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Question Number 90151 by JosephK last updated on 21/Apr/20

find the limit of lim (1/(t((√(1+t))))−(1/t) t→0

$${find}\:{the}\:{limit}\:{of}\: \\ $$$${lim}\:\:\frac{\mathrm{1}}{{t}\left(\sqrt{\mathrm{1}+{t}}\right.}−\frac{\mathrm{1}}{{t}} \\ $$$${t}\rightarrow\mathrm{0} \\ $$

Commented by abdomathmax last updated on 21/Apr/20

f(t)=(1/(t(√(1+t))))−(1/t) ⇒f(t)=((t−t(√(1+t)))/(t^2 (√(1+t)))) =((1−(√(1+t)))/(t(√(1+t)))) we have (√(1+t))∼ 1+(t/2) +(1/2)((1/2))((1/2)−1)t^2 =1+(t/2)−(t^2 /8) ⇒1−(√(1+t))∼−(t/2)+(t^2 /8) t(√(1+t))∼t(1+(t/2)−(t^2 /8))∼t+(t^2 /2) ⇒ f(t)∼((−(t/2)+(t^2 /8))/(t+(t^2 /2))) =((−(1/2)+(t/8))/(1+(t/2))) ⇒lim_(t→0) f(t)=−(1/2)

$${f}\left({t}\right)=\frac{\mathrm{1}}{{t}\sqrt{\mathrm{1}+{t}}}−\frac{\mathrm{1}}{{t}}\:\Rightarrow{f}\left({t}\right)=\frac{{t}−{t}\sqrt{\mathrm{1}+{t}}}{{t}^{\mathrm{2}} \sqrt{\mathrm{1}+{t}}} \\ $$$$=\frac{\mathrm{1}−\sqrt{\mathrm{1}+{t}}}{{t}\sqrt{\mathrm{1}+{t}}}\:\:{we}\:{have}\:\sqrt{\mathrm{1}+{t}}\sim\:\mathrm{1}+\frac{{t}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right){t}^{\mathrm{2}} \\ $$$$=\mathrm{1}+\frac{{t}}{\mathrm{2}}−\frac{{t}^{\mathrm{2}} }{\mathrm{8}}\:\Rightarrow\mathrm{1}−\sqrt{\mathrm{1}+{t}}\sim−\frac{{t}}{\mathrm{2}}+\frac{{t}^{\mathrm{2}} }{\mathrm{8}} \\ $$$${t}\sqrt{\mathrm{1}+{t}}\sim{t}\left(\mathrm{1}+\frac{{t}}{\mathrm{2}}−\frac{{t}^{\mathrm{2}} }{\mathrm{8}}\right)\sim{t}+\frac{{t}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow \\ $$$${f}\left({t}\right)\sim\frac{−\frac{{t}}{\mathrm{2}}+\frac{{t}^{\mathrm{2}} }{\mathrm{8}}}{{t}+\frac{{t}^{\mathrm{2}} }{\mathrm{2}}}\:=\frac{−\frac{\mathrm{1}}{\mathrm{2}}+\frac{{t}}{\mathrm{8}}}{\mathrm{1}+\frac{{t}}{\mathrm{2}}}\:\Rightarrow{lim}_{{t}\rightarrow\mathrm{0}} \:\:{f}\left({t}\right)=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Answered by jagoll last updated on 22/Apr/20

lim_(t→0) ((1−(√(1+t)))/(t(√(1+t)))) = lim_(t→0) ((1−(1+(t/2)))/t) lim_(t→0) ((−(t/2))/t) = −(1/2)

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

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