Question

Need help with this limit problem

lim xsin(pi/x)
x--> infinity

Answer 1

Do you know L'Hopital's rule?

Answer 2

yes but i dont know how to do it on this one

Answer 3

i am not very good with trig identities

Answer 4

First rewrite like so:\[\lim_{x\to\infty}\frac{\sin\left(\dfrac{\pi}{x}\right)}{\dfrac{1}{x}}\]
As \(x\to\infty\), you get the indeterminate expression \(\dfrac{0}{0}\). Take your derivatives.

Answer 5

No trig identities necessary here... unless you count the derivatives? I wouldn't.

Answer 6

wait...where did you get the 1/x from

Answer 7

I take reciprocals of reciprocals:\[x=\frac{1}{\frac{1}{x}}\]

Answer 8

so the x changes into 1/x

Answer 9

Or another way of looking at it: I multiply the numerator and denominator by the multiplicative inverse of \(x\).
\[\begin{align*}x\sin\frac{\pi}{x}&=\frac{x\sin\frac{\pi}{x}}{1}\\\\
&=\frac{x\sin\frac{\pi}{x}}{1}\frac{\frac{1}{x}}{\frac{1}{x}}\\\\
&=\frac{\sin\frac{\pi}{x}}{\frac{1}{x}}
\end{align*}\]

Answer 10

does that happen every time an x is in front of an identity for example xcos(pi/x)=cos(pi/x)/(1/x)

Answer 11

That's true, but what identity are you talking about? There's a difference between a trig identity and a trig expression.

Answer 12

i meant trig identity

Answer 13

No, you misunderstand my question. A trig "identity" is the statement that two trig expressions are the same. A trig expression is something like \(\sin x\) or \(\cos x\). You don't use any identities here.

Answer 14

oh okay its a trig expression

Answer 15

So do you get how to proceed with the limit?

Answer 16

can you show me how to do the derivative part?

Answer 17

i know that sin is cos the x turns into 1 but what about the pi/x and 1/x?

Answer 18

The denominator is simple power rule:
\[\frac{d}{dx}\left[\frac{1}{x}\right]=\frac{d}{dx}[x^{-1}]=-x^{-2}=-\frac{1}{x^2}\]
For the numerator, you have to use the chain rule:
\[\frac{d}{dx}\left[\sin\frac{\pi}{x}\right]=\cos\frac{\pi}{x}\cdot\frac{d}{dx}\left[\frac{\pi}{x}\right]=-\frac{\pi}{x^2}\cos\frac{\pi}{x}\]

Answer 19

oh okay i see

Answer 20

do you have time to help me with another problem i am having trouble with?

Answer 21

Maybe. Post it as another question. If I'm not too busy I'll get to it, otherwise give someone else a chance to help if they can.

Answer 22

alright

Answer 23

how do i give you points for helping?

Answer 24

Not here for the points :)

Answer 25

lol okay