Finding the Limit of x*sin(1/x) using L'Hopital's Rule

[PhysicoRaj]

Homework Statement

Evaluate: $\displaystyle\lim_{x\rightarrow0}x\sin (\frac{1}{x})$

Homework Equations

$L'H\hat{o}pital's~rule$ (?)

The Attempt at a Solution

Taking the x to denominator as $\displaystyle\lim_{x\rightarrow0}\frac{\sin(\frac{1}{x})}{\frac{1}{x}}$ and Applying L'Hopital's rule I get $$\displaystyle\lim_{x\rightarrow0}\cos {\frac{1}{x}}$$
Have I done right? How should I proceed? Thanks.

 

[Simon Bridge]

To apply L'Hopitals rule, you need to put the expression in form so you get 0/0 or infinity/infinity when you apply the limit directly.

You did not manage to do that - applying the limit directly to the rearranged equation shows form "oscillating/infinity".

Do you know the sandwich theorem?
http://www.math.washington.edu/~conroy/general/sin1overx/

 

[HallsofIvy]

$-1\le sin(1/x)\le 1$ so $-x\le x sin(1/x)\le x$.

 

[PhysicoRaj]

Thanks Simon and HallsofIvy, I understand like this: The function xsin(1/x) oscillates b/n x and -x, and as x approaches zero (left or from right), x and -x approach zero. Hence the function, which is sandwiched between these two MUST also approach zero. (I didn't know the sandwich theorem, I learned it now. Thanks )
This is one more evidence that L'Hopital's rule is not the king. Maybe I needed a more graphical, more practical view.

 

[Simon Bridge]

L'Hopital's rule is not the sole monarch of differential calculus, no. Before I started on PF I didn't even know the approach had a special name... none of my teachers formally taught it but we all knew the principle as part of the general behavior of limits.

As you advance you will learn many other rules and theorems. Together they form a "toolkit" that will help you work out what to do with specific problems. When you get good, you will end up facing problems where there is no known solution and you are the one who has to come up with the method. Enjoy.

 

[PhysicoRaj]

L'Hopital's rule is not the sole monarch of differential calculus, no. Before I started on PF I didn't even know the approach had a special name... none of my teachers formally taught it but we all knew the principle as part of the general behavior of limits.

As you advance you will learn many other rules and theorems. Together they form a "toolkit" that will help you work out what to do with specific problems. When you get good, you will end up facing problems where there is no known solution and you are the one who has to come up with the method. Enjoy.

Then I would really like it. :!) Maybe this is why 'math' is the 'science' of numbers.

 

[Simon Bridge]

That's the spirit - enjoy :)