Limit of (sin x)/x

How to use the limit of (sin x)/x to solve the given limit with a trigonometric function: formula, 4 examples, and their solutions.

Formula

Formula

The limit of (sin x)/x as x → 0 is
1.

Example 1

Example

Solution

First write the limit part and sin 4x.

The inner part of sin 4x is 4x.

So write the same 4x
in the denominator.

The denominator of the given expression is x.

But you wrote 4x.

So, to undo the denominator 4,
multiply 4.

So (sin 4x)/x = (sin 4x)/4x⋅4.

As x → 0,
(sin 4x)/4x → 1
and write the constant 4.

1⋅4 = 4

So 4 is the answer.

Example 2

Example

Solution

First write the limit part and sin (sin x).

The inner part of sin (sin x) is sin x.

So write the same sin x
in the denominator.

To undo sin x in the denominator,
write sin x in the numerator.

Write the denominator x.

So (sin (sin x))/x = [(sin (sin x))/(sin x)]⋅[(sin x)/x].

As x → 0,
(sin (sin x))/(sin x) → 1
and (sin x)/x → 1.

1⋅1 = 1

So 1 is the answer.

Example 3

Example

Solution

tan x = (sin x)/(cos x)

Quotient Identity

Switch cos x and x.

As x → 0,
(sin x)/x → 1
and cos x → 1.

Cosine Values of Commonly Used Angles

1⋅[1/1] = 1

So 1 is the answer.

Example 4

Example

Solution

To change (1 - cos x),
multiply the conjugate of (1 - cos x), (1 + cos x)
to both of the numerator and the denominator.

(1 - cos x)(1 + cos x)
= 12 - cos2 x
= 1 - cos2 x

Product of a Sum and a Difference: (a + b)(a - b)

1 - cos2 x = sin2 x

Pythagorean Identity

(sin2 x)/x2 = [(sin x)/x]2

As x → 0,
[(sin x)/x]2 → 12,
and cos x → 1.

12⋅[1/(1 + 1)]
= 1⋅1/2
= 1/2

So 1/2 is the answer.