Question Number 214719 by RoseAli last updated on 18/Dec/24
Answered by som(math1967) last updated on 18/Dec/24
$$\:{let}\:{x}−\mathrm{1}={a} \\ $$$$\:{x}\rightarrow\mathrm{1}\Rightarrow\left({x}−\mathrm{1}\right)\rightarrow\mathrm{0}\Rightarrow{a}\rightarrow\mathrm{0} \\ $$$$\:\underset{{a}\rightarrow\mathrm{0}} {{lim}}\frac{{sina}}{{a}}\:=\mathrm{1} \\ $$
Answered by sayedahmad last updated on 18/Dec/24
$$\mathrm{1} \\ $$
Answered by MathematicalUser2357 last updated on 18/Dec/24
$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{sin}\left({x}−\mathrm{1}\right)}{{x}−\mathrm{1}} \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{simplify}\:{x}\rightarrow\mathrm{1}\:\mathrm{to}\:{x}−\mathrm{1}\rightarrow\mathrm{0}. \\ $$$$\mathrm{Let}\:{a}={x}−\mathrm{1},\:\mathrm{Use}\:\mathrm{limit}\:\mathrm{idenity} \\ $$$$=\underset{{a}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{a}}{{a}}=\mathrm{1} \\ $$
Answered by MrGaster last updated on 19/Dec/24
$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}−\mathrm{1}\right)}{{x}−\mathrm{1}}\underbrace{\rightarrow\underset{{t}\rightarrow\mathrm{0}} {}\mathrm{lim}}\left(\frac{\mathrm{sin}\left({t}\right)}{{t}}\right)=\mathrm{1}. \\ $$