Question

Limita cand x tinde la infinit din x* (sin 1/x)=?

Answer 1

[tex]\lim _{x\to \infty \:}\left(x\sin \left(\frac{1}{x}\right)\right) a\cdot \:b=\frac{a}{\frac{1}{b}},\:b\ne 0 \rightarrow: L.Hopital =\lim _{x\to \infty \:}\left(\frac{\sin \left(\frac{1}{x}\right)}{\frac{1}{x}}\right)=\lim _{x\to \infty \:}\left(\frac{-\frac{\cos \left(\frac{1}{x}\right)}{x^2}}{-\frac{1}{x^2}}\right) Aplicam regula lui Chain: \mathrm{daca}\:\lim _{u\:\to \:b}\:f\left(u\right)=L,\:\mathrm{si}\:\lim _{x\:\to \:a}g\left(x\right)=b [/tex]
[tex]\mathrm{si}\:f\left(x\right)\:\mathrm{este\:continua\:pentru}\:x=b \mathrm{Atunci:}\:\lim _{x\:\to \:a}\:f\left(g\left(x\right)\right)=L g\left(x\right)=\frac{1}{x},\:f\left(u\right)=\cos \left(u\right) \:f\left(u\right)=\cos \left(u\right) \lim _{x\to \infty \:}\left(\frac{1}{x}\right) \lim _{u\to \:0}\left(\cos \left(u\right)\right) =\cos \left(0\right) = 1 Deci rezultatul final este \boxed 1[/tex]