Why is the limit of sin(x)/x equal to 1 as x tends to 0?

Question

In our HSC textbook this is just given as a standard result and we use it everywhere. But I don't understand why it equals 1 when both top and bottom go to 0.

Kavita Rao · Accepted Answer

It is a 0/0 indeterminate form, so you cannot just substitute. The result comes from the squeeze (sandwich) theorem. For small positive x, comparing areas of a triangle, a sector and a larger triangle in a unit circle gives sin x < x < tan x. Dividing through by sin x and taking reciprocals leads to cos x < sin(x)/x < 1. As x tends to 0, cos x tends to 1, so sin(x)/x is squeezed between two quantities both approaching 1, forcing the limit to be 1. Remember x must be in radians for this to hold. Many related limits like (1 − cos x)/x² = 1/2 are derived from this one.

Why is the limit of sin(x)/x equal to 1 as x tends to 0?

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