Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 115541 by bobhans last updated on 26/Sep/20

lim_(x→a) (2−(x/a))^(tan (((πx)/(2a)))) =?

$$\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\left(\mathrm{2}−\frac{{x}}{{a}}\right)^{\mathrm{tan}\:\left(\frac{\pi{x}}{\mathrm{2}{a}}\right)} =? \\ $$

Answered by bemath last updated on 26/Sep/20

L = lim_(x→a) (2−(x/a))^(tan (((πx)/(2a)))) ln L=lim_(x→a) ((ln (2−(x/a)))/(cot (((πx)/(2a))))) ln L = lim_(x→a) ((−(1/a))/(2−(x/a))) . (1/((−(π/(2a)) csc^2 (((πx)/a))))) ln L = lim_(x→a) ((2.sin^2 (((πx)/(2a))))/(π (((2a−x)/a)))) = (2/π) L = e^(2/π) = (e^2 )^(1/(π ))

$${L}\:=\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\left(\mathrm{2}−\frac{{x}}{{a}}\right)^{\mathrm{tan}\:\left(\frac{\pi{x}}{\mathrm{2}{a}}\right)} \\ $$$$\mathrm{ln}\:{L}=\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{2}−\frac{{x}}{{a}}\right)}{\mathrm{cot}\:\left(\frac{\pi{x}}{\mathrm{2}{a}}\right)} \\ $$$$\mathrm{ln}\:{L}\:=\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{−\frac{\mathrm{1}}{{a}}}{\mathrm{2}−\frac{{x}}{{a}}}\:.\:\frac{\mathrm{1}}{\left(−\frac{\pi}{\mathrm{2}{a}}\:\mathrm{csc}^{\mathrm{2}} \left(\frac{\pi{x}}{{a}}\right)\right)} \\ $$$$\mathrm{ln}\:{L}\:=\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{\mathrm{2}.\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\pi{x}}{\mathrm{2}{a}}\right)}{\pi\:\left(\frac{\mathrm{2}{a}−{x}}{{a}}\right)}\:=\:\frac{\mathrm{2}}{\pi} \\ $$$${L}\:=\:{e}^{\frac{\mathrm{2}}{\pi}} \:=\:\sqrt[{\pi\:}]{{e}^{\mathrm{2}} } \\ $$

Answered by Dwaipayan Shikari last updated on 26/Sep/20

lim_(x→a) (1+1−(x/a))^(tan(((πx)/(2a)))) =y lim_(x→a) tan(((πx)/(2a)))log(1+1−(x/a))=logy (1−(x/a))tan(((πx)/(2a)))=logy (1−(x/a))sin(((πx)/(2a))).(1/(cos(((πx)/(2a)))))=logy (1−(x/a))(1/(sin((π/2)−((πx)/(2a)))))=logy (((1−(x/a)))/((π/2)(1−(x/a))))=logy y=e^(2/π)

$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\left(\mathrm{1}+\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)^{\mathrm{tan}\left(\frac{\pi\mathrm{x}}{\mathrm{2a}}\right)} =\mathrm{y} \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}tan}\left(\frac{\pi\mathrm{x}}{\mathrm{2a}}\right)\mathrm{log}\left(\mathrm{1}+\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)=\mathrm{logy} \\ $$$$\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)\mathrm{tan}\left(\frac{\pi\mathrm{x}}{\mathrm{2a}}\right)=\mathrm{logy} \\ $$$$\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)\mathrm{sin}\left(\frac{\pi\mathrm{x}}{\mathrm{2a}}\right).\frac{\mathrm{1}}{\mathrm{cos}\left(\frac{\pi\mathrm{x}}{\mathrm{2a}}\right)}=\mathrm{logy} \\ $$$$\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\pi}{\mathrm{2}}−\frac{\pi\mathrm{x}}{\mathrm{2a}}\right)}=\mathrm{logy} \\ $$$$\frac{\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)}{\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{a}}\right)}=\mathrm{logy} \\ $$$$\mathrm{y}=\mathrm{e}^{\frac{\mathrm{2}}{\pi}} \\ $$$$ \\ $$

Answered by mnjuly1970 last updated on 26/Sep/20

solution: 1−(x/a) =t {_(t→0) ^(x→a) lim_(t→0) (1+t)^(tan[(π/2)(1−t)]) =lim_(t→0 ) (1+t)^(cot((π/2)t)) =_(t→0) ^( sin((π/2)t)∽(π/2)t) lim_(t→0) [((1+t)^(2/π) )^(1/t) ]^(cos((π/2)t)) =_(t→0) ^(cos((π/2)t)→1) =e^(2/π) ✓ m.n.july.1970

$$\:{solution}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}−\frac{{x}}{{a}}\:={t}\:\left\{_{{t}\rightarrow\mathrm{0}} ^{{x}\rightarrow{a}} \right. \\ $$$$\:\:{lim}_{{t}\rightarrow\mathrm{0}} \:\left(\mathrm{1}+{t}\right)^{{tan}\left[\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−{t}\right)\right]} \\ $$$$={lim}_{{t}\rightarrow\mathrm{0}\:} \left(\mathrm{1}+{t}\right)^{{cot}\left(\frac{\pi}{\mathrm{2}}{t}\right)} \underset{{t}\rightarrow\mathrm{0}} {\overset{\:\:\:\:\:\:{sin}\left(\frac{\pi}{\mathrm{2}}{t}\right)\backsim\frac{\pi}{\mathrm{2}}{t}} {=}} \\ $$$${lim}_{{t}\rightarrow\mathrm{0}} \left[\left(\left(\mathrm{1}+{t}\right)^{\frac{\mathrm{2}}{\pi}} \right)^{\frac{\mathrm{1}}{{t}}} \right]^{{cos}\left(\frac{\pi}{\mathrm{2}}{t}\right)} \underset{{t}\rightarrow\mathrm{0}} {\overset{{cos}\left(\frac{\pi}{\mathrm{2}}{t}\right)\rightarrow\mathrm{1}} {=}} \\ $$$$\:\:\:\:\:={e}^{\frac{\mathrm{2}}{\pi}} \:\checkmark \\ $$$$\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com