How do I find the limit as $x$ $x$ approaches infinity of $x sin (\frac{1}{x})$ $xsin(1/x)$ ?

Question

6192 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-26T23:28:32

By l'Hopital's Rule,

$lim_{x to infty}x sin(1/x)=1$

Let us look at some details.

$lim_{x to infty}x sin(1/x)$

by rewriting a bit,

$=lim_{x to infty}{sin(1/x)}/{1/x}$

by l'Hopital's Rule,

$=lim_{x to infty}{cos(1/x)cdot(-1/x^2)}/{-1/x^2}$

by cancelling out $-1/x^2$ ,

$=lim_{x to infty}cos(1/x)=cos(0)=1$

How do I find the limit as xxx approaches infinity of xsin(1/x)xsin(1x)xsin(1/x)?