The limit of cosx/x as x approaches 0 **does NOT** exist. That is, lim_{x→0} (cos x)/x is undefined.

$\lim\limits_{x \to 0} \dfrac{\cos x}{x}$ = NOT exist.

Let us now show that the limit of (cos x)/x when x tends to 0 does not exist.

## Lim_{x→0} (cos x)/x

**Question: **What is the limit of cosx/x when x tends to 0?

**Solution:**

To show the limit of cosx/x does not exist, we will tend to 0 in two different paths to obtain different limits.

Take x_{n} = 1/n and x_{n}= -1/n.

Both tend to 0 when n→∞.

Observe that $\dfrac{1}{n} \in [0,\frac{\pi}{4}]$, so cos(x_{n}) = cos(1/n) ≥ 1/2.

Therefore,

$\dfrac{\cos x_n}{x_n} \geq \dfrac{1}{2x_n} \geq \dfrac{n}{2}$ **…(I)**

Similarly, for x_{n}= -1/n, we can show that

$\dfrac{\cos x_n}{x_n} \leq \dfrac{1}{2x_n} \leq \dfrac{-n}{2}$ **…(II)**

From **(I)** and **(II)**, we see that $\dfrac{\cos x_n}{x_n}$ approaches to two different limits when x_{n}→0, that is n→∞.

So the limit of cosx/x does not exist. This is proved by taking the limit x→0 in two different paths and obtain two different limits.

**Read Also:** Limit of cosx/x when x→∞

lim_{x→0} sin(√x)/x | lim_{x→0} sin(x^{2})/x

lim_{x→0} tanx/x | lim_{x→0}(cosx-1)/x

## FAQs

### Q1: What is the limit of cosx/x when x tends to 0?

Answer: The limit of cosx/x as x tends to 0 does NOT exist.