Two limits:
1) PQ approaches the arc s: The arc s is caught between PQ and PC +CQ < PB. Drive θ to zero. The differences between the three lengths can be made as small as we wish (smaller than any finite quantity).
Result 1: In the limit, the tangent PB, arc s, and subtended chord PQ are equal. We may use them interchangeably.
2) lim sinθ/θ = 1:

Put sinθ and θ in a box together.
Pick θ or sinθ. I choose θ. Create two inequalities:
θ > A sinθ
θ < B sinθ
A, B expressions in θ and x. This is the box:
A sinθ < θ < B sinθ
Put sinθ/θ in the middle.

Shut the box: drive x to 0 (the limiting value).

The left and right sides are defined and positive for all values of x < 1. How far apart are they at x=0?
They are the same.

It is always possible to take a limit in this way (place the ratio in question at the center of an inequality, and let the bounds approach the limit). The upside and downside are the same: I will get the limiting interval I asked for. I could choose badly and get an answer like: -∞ <sinθ/θ < ∞.
Here, I found two expressions which are equal at θ=0.