Limit as x -> 0 of sin(x)/x Step by Step

Step by step

The question

Question: find the limit as x → 0 of sin x over x.

lim x → 0 sin xx =

1 Put the number in

Put 0 in place of x.

The top is 0, and the bottom is 0.

top: sin 0 = 0

bottom: 0

0/0 has no value.

It tells us nothing yet.

This is where L'Hôpital's rule helps.

form 00

2 Use L'Hôpital's rule

The rule: when we get 0/0, or infinity over infinity, we may take the derivative of the top, and the derivative of the bottom.

The new limit is the same.

lim x → a f(x)g(x) = lim x → a f′(x)g′(x)

Careful!

This is not the quotient rule.

The top gets its own derivative, and the bottom gets its own derivative.

Careful: not the quotient rule: top and bottom each alone

The derivative of the top is cos x.

The derivative of the bottom is 1.

The new limit is the same.

lim x → 0 sin xx = lim x → 0 cos x1

3 Put the number in again

Put 0 in again.

Now the bottom is not 0.

So the limit is 1.

= cos 01 = 11 = 1

4 Check

Let's check with a calculator in radians, with x very close to 0.

x = −0.01 gives about 1.

x = −0.01: sin xx ≈ 1

And x = 0.01 gives about 1.

Both are close to 1.

Right!

x = 0.01: sin xx ≈ 1

Let's look back.

First: put the number in.

Then: use L'Hôpital's rule.

Then: put the number in again.

Last: check.

Answer