Problem Analysis #calculus_23

Recorded Python call:

import math

def f(x):
    return math.tan(6*x) / (2*x)

# evaluate near 0
for h in [0.1, 0.01, 0.001, 0.0001, 0.00001]:
    print(f"h={h:.6f}, f(x) = {f(h):.10f}")

We evaluate the limit using the standard result $\lim_{u\to 0}\frac{\tan u}{u}=1$. Let $u = 6x$, then as $x\to 0$, $u\to 0$. We have:

Taking the limit as $x\to 0$:

A numerical check confirms this: for $x$ values approaching 0 (0.1, 0.01, 0.001, 0.0001, 0.00001), the function values approach 3.

Thus, the answer is ③.

$\boxed{3}$