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- anonymous

What is the limit of (x Sin (1/x)) as x becomes positive infinite?

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- anonymous

What is the limit of (x Sin (1/x)) as x becomes positive infinite?

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- anonymous

i believe the limit is 0 because as x increases, 1/x decreases to where it is approaching zero and the sin (0) is 0. and anything multiplied by 0 is 0.

- anonymous

Use a substitution u=1/x, then the limit will be
\[\lim_{u \rightarrow 0}{\sin x \over x}=1\]

- anonymous

it is 0

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- anonymous

Sorry
\[\lim_{u \rightarrow 0} {\sin u \over u}=1\]

- anonymous

limits is zero , its seen by pinching theorm

- anonymous

Anwar..i think you might have to use L'Hopital's Rule.

- anonymous

I don't think L'hopital's rule can work here. I still believe it's 1 by the substitution I used.

- anonymous

check it, sub in a big number'

- anonymous

it goes to zero

- anonymous

the larger you're number gets, the closer you are to zero because it is a fraction.

- anonymous

@thamir: When substituting \(u=1/x\), \(\sin (1/x)= \sin u\) and \(x=1/u\). Also the approaching point will be zero instead of infinity since \(1/\infty \rightarrow 0\).

- anonymous

-1+infinity

- anonymous

-x

- anonymous

wait, thats not going to work :|

- anonymous

something similar lol

- anonymous

@elecengineer: When you substitute a large number the x goes to \(\infty \) and the sin(1/x) goes to \(0\). And \( \infty \times 0 \ne0\).

- anonymous

L'Hopitals Rule does work AnwarA and you are right the answer is 1.
lim sin(1/x)/(1/x)
=lim [(-1/x^2)cos(1/x)]/(-1/x^2)
=lim cos(1/x)
=cos(0)=1

- myininaya

\[\lim_{x \rightarrow \inf}\frac{\sin(\frac{1}{x})}{\frac{1}{x}}\]
let u=1/x
x->inf, u->0
so the limit is 1

- anonymous

Thanks AnwarA for solving it. Thanks chris777 for teaching me the L'Hopitals Rule. And last but not least thanks to elecengineer for trying to solve using the squeeze theorem.
Now, I need to get the correct answer which is 1 by using the squeeze theorem. When trying I reached the same point where elecengineer reached. How can we solve it using the squeeze theorem?

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