# Does lim x →0 sin 1/x exist? also .

1. May 24, 2009

### hermy

Does lim x →0 sin 1/x exist? also.....

1. The problem statement, all variables and given/known data

does lim x →0 sin(1/x) exist?

3. The attempt at a solution

I am very new to calculus. From what i have understood till now, limit of a function at a point exists only if the left and right hand limits are equal. What would the graph of sin(1/x) look like? Would the left and right hand limits exist? Do they exist in all cases, for all functions?

2. May 24, 2009

### matt grime

Re: Does lim x →0 sin 1/x exist? also.....

try drawing it and find out.

It would be a bit dull if left and right limits always exist. Remember that limits are unique. So, you need to think about sequences of real numbers x_n tending to zero.

lim sin(y)

as y tends to infinity.

3. May 24, 2009

### kbaumen

Re: Does lim x →0 sin 1/x exist? also.....

Btw, the existence of left and right limits of a function at a certain point is necessary but not sufficient for a limit to exist at that point. Left and right limits should also be equal for a limit to exist at the same point.

4. May 24, 2009

### HallsofIvy

Staff Emeritus
Re: Does lim x →0 sin 1/x exist? also.....

In particular, think about the sequences $x_n= 1/(n\pi)$, $x_n= 2/((4n+1)\pi)$, and $x_n= 2/((4n-1)\pi)$ a n goes to infinity.

5. May 24, 2009

### Random Variable

Re: Does lim x →0 sin 1/x exist? also.....

In a real analysis class you would do something like the following:

Let $$f(x) = sin ( \frac {1}{x} )$$

Let $$s_{n}$$ be the sequence $$\frac {2}{n \pi}$$

$$\lim_{n \to \infty}s_{n} = 0$$

but $$(f(s_{n}) )$$ is the sequence 1,0,-1,0,1,0,-1,0... which obviously doesn't converge.

Therefore, $$\lim_{x \to 0} sin( \frac {1}{x})$$ does not exist.

6. May 24, 2009

### tiny-tim

Hi hermy!

matt's is the best way for general questions like this …

it shows you what the difficulty is!

7. May 25, 2009

### hermy

Re: Does lim x →0 sin 1/x exist? also.....

Could you give an example of a function where the left/right hand limit doesn't exist at a point (where the function is defined on the set of all real no.s)?

8. May 25, 2009

### Staff: Mentor

Re: Does lim x →0 sin 1/x exist? also.....

f(x) = 0 if x is rational, 1 if it is irrational.

9. May 25, 2009

### Staff: Mentor

Re: Does lim x →0 sin 1/x exist? also.....

Another example is f(x) = 1/x, x $\neq$ 0; f(0) = 0.
This function is defined for all real numbers. The left hand limit, as x approaches zero is negative infinity, while the right hand limit, as x approaches zero, is positive infinity.

10. May 25, 2009

### chiro

Re: Does lim x →0 sin 1/x exist? also.....

In calculus you need to use the idea of limits. Essentially the limit of sin x/x does equal 1 but you have to show it from both sides.

If we consider a left hand limit if sin x/x then we can verify the limit with a table of calculations (just grab a calculator and calculate f(x) sin x / x for values approaching x)

We can also consider the right hand limit also. For the right hand limit we can do the same thing by letting f(x) approach sin x/x.

Now the limit is only valid if and only if the right hand limit equals the left hand limit. So
essentially if this is the case and you find a value for it, then you can prove what that limit
is.

11. May 25, 2009

### Staff: Mentor

Re: Does lim x →0 sin 1/x exist? also.....

The original poster is looking at f(x)=sin(1/x), not g(x)=sin(x)/x.

12. May 25, 2009

### matt grime

Re: Does lim x →0 sin 1/x exist? also.....

I don't want to give the game away but.... sin(1/x) as x tends to 0 perhaps? I can trivially extend this to a function on the whole of the real line if I want to, if you're worried about it being undefined at zero:

f(x)=sin(1/x) x=/=0
f(0)=0 (or anything else you want)