The behavior of the functions sin(1/x) and x sin(1/x) when x is near zero are worth noting.

Below are plots of sin(1/x) for small positive x.

We can see that as *x* gets closer to zero, the function keeps
wobbling (or *oscillating*) back and forth between -1 and 1.

In fact, sin(1/x) wobbles between -1 and 1 an infinite number of times
between 0 and any positive *x* value, no matter how small.

To see this, consider that sin(x) is equal to zero at every multiple of pi,
and it wobbles between 0 and 1 or -1 between each multiple.
Hence, sin(1/x) will be zero at every x = 1/(pi *k*), where *k* is a
positive integer. In between each consecutive pair of these values, sin(1/x)
wobbles from 0, to -1, to 1 and back to 0.

There are an infinite number of these pairs, and they are all between 0 and 1/pi.
What's more, there are only finitely many between any positive *x* value and
1/pi, so there must be infinitely many between that *x* and 0.

We can conclude that as *x* approaches 0 from the right, the function
sin(1/x) does not settle down on any value *L*, and so the limit as
*x* approaches 0 from the right does not exist.

Now, the function x sin(1/x) is a somewhat different story.
Since *x* approaches zero as *x* approaches zero,
multiplying sin(1/x) by it will result in another quantity that approaches zero.
Below is some visual evidence. The yellow lines are y=x and y=-x, while the
blue curve is x sin(1/x):

This is an example of what's known as the Sandwich Theorem.

The Sandwich Theorem says that

if g(x) ≤ f(x) ≤ h(x), and

g(x) and h(x) both approach L as x approaches *a*,

then f(x) must also approach L as x approaches *a*.

In this case, we know that, since -1 ≤ sin(1/x) ≤ 1, we can conclude that -x ≤ x sin(1/x) ≤ x for positive values of x. Then, since x and -x both approach 0 as x approaches 0 from the right, so must x sin(1/x).

You can make a similar argument from the left, and conclude that
the limit as *x* approaches 0 of *x* sin(1/*x*) is 0.
Here's a picture that shows the goings on to the left of zero.

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