Why is the limit of sin x over x equal to 1?

Why direct substitution does not answer it

Substituting zero produces 0/0. That expression is not the value of the limit; it is a warning that two quantities are approaching zero and their relative rates still need to be compared.

The limit asks about nearby nonzero inputs. The function can be undefined at zero and still approach a perfectly definite value there.

The geometric squeeze

On the unit circle, for a small positive angle x in radians, the vertical segment has length sin x, the arc has length x, and the tangent segment has length tan x. Their areas give sin x < x < tan x.

After dividing carefully and using symmetry for negative x, cos x is less than sin x divided by x, which is less than 1. Both outside expressions approach 1.

Why radians matter

The arc length on the unit circle equals the angle only in radians. In degrees, the limit would carry a conversion factor. Calculus formulas look clean because radians build that factor into the definition of angle.

Worked example

Common mistakes

• Treating 0/0 as an answer instead of an indeterminate form.
• Using degrees without the conversion factor.
• Applying the fact when the inner expression does not approach zero.

Keep these ideas

• A missing function value does not prevent a limit.
• The proof is a squeeze built from unit-circle geometry.
• Recognize and manufacture the sin u / u pattern.