DAY 2002A Limits Practice Test 2026 – Complete Exam Preparation

The main idea is that near zero, sine behaves like its input when the input is measured in radians. So sin x / x should approach 1 as x → 0.

A standard way to see this uses a geometric squeeze on the unit circle. For small positive x, sin x ≤ x ≤ tan x = sin x / cos x. Dividing by sin x (which is positive near 0) gives 1 ≤ x/ sin x ≤ 1/ cos x. Taking reciprocals yields cos x ≤ sin x/ x ≤ 1. As x → 0, both cos x and 1 approach 1, so by the Squeeze Theorem, sin x/ x → 1.

Alternatively, you can view sin x as x minus a higher-order term, sin x = x − x^3/6 + ..., which gives sin x/ x = 1 − x^2/6 + ..., tending to 1.

Thus the limit is 1. The other options would imply different behavior that doesn’t match the sine function’s local linearity in radians.