# What are the asymptote(s) and hole(s), if any, of  f(x) = xsin(1/x)?

Feb 4, 2018

Refer below.

#### Explanation:

Well, there is obviously a hole at $x = 0$, since division by $0$ is not possible.

We can graph the function:
graph{xsin(1/x) [-10, 10, -5, 5]}

There are no other asymptotes or holes.

Feb 4, 2018

$f \left(x\right)$ has a hole (removable discontinuity) at $x = 0$.

It also has a horizontal asymptote $y = 1$.

It has no vertical or slant asymptotes.

#### Explanation:

Given:

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

I will use a few of properties of $\sin \left(t\right)$, namely:

• $\left\mid \sin t \right\mid \le 1 \text{ }$ for all real values of $t$.

• ${\lim}_{t \to 0} \sin \frac{t}{t} = 1$

• $\sin \left(- t\right) = - \sin \left(t\right) \text{ }$ for all values of $t$.

First note that $f \left(x\right)$ is an even function:

$f \left(- x\right) = \left(- x\right) \sin \left(\frac{1}{- x}\right) = \left(- x\right) \left(- \sin \left(\frac{1}{x}\right)\right) = x \sin \left(\frac{1}{x}\right) = f \left(x\right)$

We find:

$\left\mid x \sin \left(\frac{1}{x}\right) \right\mid = \left\mid x \right\mid \left\mid \sin \left(\frac{1}{x}\right) \right\mid \le \left\mid x \right\mid$

So:

$0 \le {\lim}_{x \to 0 +} \left\mid x \sin \left(\frac{1}{x}\right) \right\mid \le {\lim}_{x \to 0 +} \left\mid x \right\mid = 0$

Since this is $0$, so is ${\lim}_{x \to 0 +} x \sin \left(\frac{1}{x}\right)$

Also, since $f \left(x\right)$ is even:

${\lim}_{x \to {0}^{-}} x \sin \left(\frac{1}{x}\right) = {\lim}_{x \to {0}^{+}} x \sin \left(\frac{1}{x}\right) = 0$

Note that $f \left(0\right)$ is undefined, since it involves division by $0$, but both left and right limits exist and agree at $x = 0$, so it has a hole (removable discontinuity) there.

We also find:

${\lim}_{x \to \infty} x \sin \left(\frac{1}{x}\right) = {\lim}_{t \to {0}^{+}} \sin \frac{t}{t} = 1$

Similarly:

${\lim}_{x \to - \infty} x \sin \left(\frac{1}{x}\right) = {\lim}_{t \to {0}^{-}} \sin \frac{t}{t} = 1$

So $f \left(x\right)$ has a horizontal asymptote $y = 1$

graph{x sin(1/x) [-2.5, 2.5, -1.25, 1.25]}