Limit of cosx/x as x approaches 0 | Lim x→0 cosx/x

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→∞

Limit of sinx/x when x→∞

Limit of sinx/x when x→0

lim_x→0 sin(√x)/x | lim_x→0 sin(x²)/x

lim_x→0 tanx/x | lim_x→0(cosx-1)/x

FAQs