Limit of sin(x)/x near 0

Near $0$, the Taylor series gives $\sin x = x - x^3/6 + x^5/120 - \ldots$. Dividing by $x$ yields $1 - x^2/6 + \ldots$, which approaches $1$ as $x\to 0$. This is the cornerstone limit for differentiating $\sin x$ from first principles.

Slide the probe $h$ closer and closer to $0$. The lab reports $\sin(h)/h$, the error $1 - \sin(h)/h$ and the next series term $-h^2/6$. At $h = 0.1$, $\sin(h)/h \approx 0.99833$, exactly $1 - 0.01/6$.

The expansion $\sin x = x - x^3/6 + \ldots$ requires $x$ in radians. If you plug in degrees, $\sin x$ grows much slower per unit $x$, and the limit becomes $\pi/180$, not $1$. Many students lose marks by computing $\sin(30^\circ)/30 = 0.0167$ and panicking. Radians always for limits.

Small-angle approximation: pendulum period derivation uses $\sin\theta \approx \theta$ for small $\theta$, valid because $\sin\theta/\theta\to 1$. Same idea underpins fringe spacing in Young's double-slit experiment and pendulum SHM. The lab gives you confidence to make the approximation in physics MCQs.