Limit of $x \sin\left(\frac{1}{x}\right)$ as $x \to 0$

How to work it out

1. 1

   We want what x * sin(1 / x) approaches as x → 0.

   $$\lim_{x \to 0} x \sin\left(\frac{1}{x}\right)$$
2. 2

   Direct substitution gives an indeterminate form: 0·bounded. It does not tell us the limit.

3. 3

   Method: squeeze: |x sin(1/x)| ≤ |x| → 0.

4. 4

   Result:

   $$\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0$$

Values of x * sin(1 / x) as x → 0 (computed)

Questions

More well-known limits

• $\frac{\sin\left(x\right)}{x}$ as $x \to 0$
• $\frac{\tan\left(x\right)}{x}$ as $x \to 0$
• $\frac{1 - \cos\left(x\right)}{x^{2}}$ as $x \to 0$
• $\frac{1 - \cos\left(x\right)}{x}$ as $x \to 0$
• $\frac{e^{x} - 1}{x}$ as $x \to 0$
• $\frac{\ln\left(1 + x\right)}{x}$ as $x \to 0$
• $\frac{\arctan\left(x\right)}{x}$ as $x \to 0$
• $\frac{\arcsin\left(x\right)}{x}$ as $x \to 0$