Question Number 155353 by zakirullah last updated on 29/Sep/21
$$\boldsymbol{{show}}\:\boldsymbol{{that}}\:\:\boldsymbol{{lim}}_{\boldsymbol{{x}}\rightarrow\mathrm{0}} \frac{\boldsymbol{{sinx}}}{\boldsymbol{{x}}}\:=\:\mathrm{1} \\ $$
Answered by puissant last updated on 29/Sep/21
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}}{{x}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}−{sin}\mathrm{0}}{{x}−\mathrm{0}}={f}'\left(\mathrm{0}\right) \\ $$$${f}\left({x}\right)={sinx}\rightarrow{f}'\left({x}\right)={cosx} \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}}{{x}}={f}'\left(\mathrm{0}\right)={cos}\left(\mathrm{0}\right)=\mathrm{1}.. \\ $$
Commented by zakirullah last updated on 29/Sep/21
$$\mathrm{sir}\:\mathrm{nice}\:\mathrm{but}\:\mathrm{very}\:\mathrm{difficult}\:\mathrm{to}\:\mathrm{understand}\:? \\ $$$$\mathrm{because}\:\mathrm{some}\:\mathrm{steps}\:\mathrm{are}\:\mathrm{skiped}. \\ $$$$\mathrm{please}\:\mathrm{do}\:\mathrm{this}\:\mathrm{very}\:\mathrm{long}\:\mathrm{process}\:\left(\mathrm{simplification}\right) \\ $$
Commented by TheHoneyCat last updated on 29/Sep/21
This answer might not be perfectly written, but I can assure you it contains all the steps, as long as you know that dsin(x)/dx is cos... But I'm pretty sure it is not what you asked for.
So do study it if you need, but know that there are no "less difficult" solutions.
Commented by zakirullah last updated on 30/Sep/21
$$\mathrm{ok}\:\mathrm{sir}\:\mathrm{i}\:\mathrm{am}\:\mathrm{understand}. \\ $$$$\mathrm{thanks}\:\mathrm{alot}. \\ $$$$\mathrm{but}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sinx}}{\mathrm{x}}\:=\:\frac{\mathrm{sinx}−\mathrm{sin0}}{\mathrm{x}−\mathrm{0}}\:=\:\mathrm{f}'\left(\mathrm{0}\right) \\ $$$$\mathrm{oR}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sinx}.\mathrm{sin0}}{\mathrm{x}.\mathrm{o}}\:=\:\mathrm{f}'\left(\mathrm{0}\right) \\ $$$$\mathrm{which}\:\mathrm{one}\:\mathrm{is}\:\mathrm{the}\:\mathrm{most}\:\mathrm{correct}. \\ $$
Answered by physicstutes last updated on 02/Oct/21
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{x}}{{x}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}}{\mathrm{1}}\:=\:\mathrm{cos}\left(\mathrm{0}\right)=\mathrm{1} \\ $$