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

How to work it out

1. 1

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

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

   Direct substitution is impossible, and the values do not settle.

3. 3

   Method: the function oscillates between −1 and 1 infinitely often near 0, so it settles on no value.

4. 4

   The table shows values jumping between about −1 and 1, so there is no limit.

   $$\lim_{x \to 0} \sin\left(\frac{1}{x}\right) = \text{does not exist}$$

Values of 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 $x \to 0$