# Limit of sin(1/x) as x approaches 0

The limit of sin(1/x) as x approaches 0 does not exist, that is, limx0 sin(1/x) is undefined. The limit formula of sin(1/x) when x→0 is given below:

limx0 sin(1/x) = undefined.

Let us learn how to find the limit of sin(1/x) when x approaches 0.

## Find the Limit of sin(1/x) when x→0

Answer: The function sin(1/x) has no defined limit when x→0.

Proof:

We will use the following theorem in order to prove the limit formula limx0 sin(1/x) = undefined.

Theorem: If limxc f(x) = L, then for every sequence {xn} converging to c the limit limn f(xn) always equals to L, that is, the limit will be unique.

Let f(x) = sin(1/x).

To show limx f(x) does not exist, consider two sequences {xn} and {yn}, both converging to 0, but f(xn) and f(yn) converge to different limits. Take

xn = 1/nπ

yn = $\dfrac{1}{(4n+1)\pi/2}$

Note that sin(nπ)=0 and sin (4n+1)π/2 = 1. Also, observe that xn, yn →0 when x tends to ∞. Now we calculate

limn f(xn) = limn sin(nπ) = 0.

On the other hand,

limn f(yn) = limn sin (4n+1)π/2 = 1.

Thus we deduce that both sequences f(xn) and f(yn) converge to different limits. Hence by the above theorem the limit of sin(1/x) when x→0 does not exist.

Epsilon – delta definition of limit

Sum rule of limits

Product rule of limits

Quotient rule of limits

## FAQs

### Q1: What is the limit of sin(1/x) as x approaches 0.

Answer: The limit of sin(1/x) as x approaches 0 is not defined.