Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 62435 by mathsolverby Abdo last updated on 21/Jun/19

calculate lim_(x→0) (((1+x)^(sinx) −1)/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\left(\mathrm{1}+{x}\right)^{{sinx}} −\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$

Commented by Smail last updated on 22/Jun/19

(1+x)^(sinx) =e^(ln((1+x)^(sinx) )) =e^(sinx×ln(1+x)) ln(1+x)∼x when x is near 0 sinx∼x when x is near 0 so (1+x)^(sinx) ∼e^(x×x) ∼e^x^2 e^x ∼1+x when x near 0 so (1+x)^(sinx) ∼1+x^2 lim_(x→0) (((1+x)^(sinx) −1)/x^2 )=lim_(x→0) ((1+x^2 −1)/x^2 ) =1

$$\left(\mathrm{1}+{x}\right)^{{sinx}} ={e}^{{ln}\left(\left(\mathrm{1}+{x}\right)^{{sinx}} \right)} ={e}^{{sinx}×{ln}\left(\mathrm{1}+{x}\right)} \\ $$$${ln}\left(\mathrm{1}+{x}\right)\sim{x}\:\:{when}\:{x}\:{is}\:{near}\:\mathrm{0} \\ $$$${sinx}\sim{x}\:\:{when}\:\:{x}\:{is}\:{near}\:\mathrm{0} \\ $$$${so}\:\left(\mathrm{1}+{x}\right)^{{sinx}} \sim{e}^{{x}×{x}} \sim{e}^{{x}^{\mathrm{2}} } \\ $$$${e}^{{x}} \sim\mathrm{1}+{x}\:{when}\:{x}\:{near}\:\mathrm{0} \\ $$$${so}\:\:\left(\mathrm{1}+{x}\right)^{{sinx}} \sim\mathrm{1}+{x}^{\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\left(\mathrm{1}+{x}\right)^{{sinx}} −\mathrm{1}}{{x}^{\mathrm{2}} }=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{1}+{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$=\mathrm{1} \\ $$

Commented by mathmax by abdo last updated on 23/Jun/19

let use hospital theorem u(x) =(1+x)^(sinx) −1 ⇒u(x) =e^(sinxln(1+x)) −1 ⇒ u^′ (x) =(cosx ln(1+x) +((sinx)/(1+x)))e^(sinxln(1+x)) ⇒ u^((2)) (x) ={ −sinxln(1+x)+((cosx)/(1+x)) +(((1+x)cosx−sinx)/((1+x)^2 )))e^(sinxln(1+x)) +(cosx ln(1+x)+((sinx)/(1+x)))^2 e^(sinxln(1+x)) ⇒lim_(x→0) u^((2)) (x)= 2 v(x) =x^2 ⇒g^′ (x) =2x and v^((2)) (x) =2 =lim_(x→0) g^((2)) (x) ⇒ lim_(x→0) (((1+x)^(sinx) −1)/x^2 ) =lim_(x→0) ((u^((2)) (x))/(v^((2)) (x))) =(2/2) =1 .

$${let}\:{use}\:{hospital}\:{theorem}\:\:{u}\left({x}\right)\:=\left(\mathrm{1}+{x}\right)^{{sinx}} −\mathrm{1}\:\Rightarrow{u}\left({x}\right)\:={e}^{{sinxln}\left(\mathrm{1}+{x}\right)} −\mathrm{1}\:\Rightarrow \\ $$$${u}^{'} \left({x}\right)\:=\left({cosx}\:{ln}\left(\mathrm{1}+{x}\right)\:+\frac{{sinx}}{\mathrm{1}+{x}}\right){e}^{{sinxln}\left(\mathrm{1}+{x}\right)} \:\Rightarrow \\ $$$${u}^{\left(\mathrm{2}\right)} \left({x}\right)\:=\left\{\:−{sinxln}\left(\mathrm{1}+{x}\right)+\frac{{cosx}}{\mathrm{1}+{x}}\:+\frac{\left(\mathrm{1}+{x}\right){cosx}−{sinx}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }\right){e}^{{sinxln}\left(\mathrm{1}+{x}\right)} \\ $$$$+\left({cosx}\:{ln}\left(\mathrm{1}+{x}\right)+\frac{{sinx}}{\mathrm{1}+{x}}\right)^{\mathrm{2}} \:{e}^{{sinxln}\left(\mathrm{1}+{x}\right)} \:\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}} \:{u}^{\left(\mathrm{2}\right)} \left({x}\right)=\:\mathrm{2} \\ $$$${v}\left({x}\right)\:={x}^{\mathrm{2}} \:\Rightarrow{g}^{'} \left({x}\right)\:=\mathrm{2}{x}\:{and}\:{v}^{\left(\mathrm{2}\right)} \left({x}\right)\:=\mathrm{2}\:={lim}_{{x}\rightarrow\mathrm{0}} {g}^{\left(\mathrm{2}\right)} \left({x}\right)\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\left(\mathrm{1}+{x}\right)^{{sinx}} −\mathrm{1}}{{x}^{\mathrm{2}} }\:={lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{{u}^{\left(\mathrm{2}\right)} \left({x}\right)}{{v}^{\left(\mathrm{2}\right)} \left({x}\right)}\:=\frac{\mathrm{2}}{\mathrm{2}}\:=\mathrm{1}\:. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com