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Calculus IDerivatives

Trig derivatives, from limits to motion.

See why the special trig limits work, build the derivative rules, then practise every example with your own answers checked.

Trig limits · Step 1 of 8

Trap a wild function between two tame ones

x² sin(3π / (x(x + 1)))±x² (the walls)
−0.9−0.6−0.300.30.60.9−0.9−0.6−0.300.30.60.9
Some functions wiggle too much to plug in or simplify. If g(x) is always between f(x) and h(x) near a, and both f and h go to the same limit L, then g has nowhere to go: its limit is L too.
Try it: Zoom in toward x = 0. The walls close in, and the wiggly curve is forced to 0 with them.
Function
Zoom window±0.9
widetight
Window
±0.9
Walls at the edge
±0.81
Limit
0
−1 ≤ sin(3π / (x(x + 1))) ≤ 1
Sine and cosine always stay between −1 and 1.
−x² ≤ x² sin(3π / (x(x + 1))) ≤ x²
Multiply by x², which is never negative, so the inequality signs stay the same.
limx → 0 (−x²) = 0 = limx → 0 x²  ⇒  limx → 0 x² sin(3π / (x(x + 1))) = 0
Both walls go to 0, so the function trapped between them does too.

Need a hand with trig limits or derivatives?

Bring your practice questions. We work through them together until every type feels routine.

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