State the fundamental sine limit for angles measured in radians.

What is \(\lim_{x\to0}\frac{\tan x}{x}\)?

Evaluate \(\lim_{x\to0}\frac{\sin(4x)}{4x}\).

Explain why \(\lim_{x\to0}\frac{1-\cos x}{x}=0\) using the identity \(1-\cos x=\frac{\sin^2x}{1+\cos x}\).

Evaluate \(\lim_{x\to0}\frac{\sin(5x)}{x}\).

Evaluate \(\lim_{x\to0}\frac{\sin(3x)}{\sin(7x)}\).

Evaluate \(\lim_{x\to0}\frac{\tan(2x)}{x}\).

Evaluate \(\lim_{x\to0}\frac{1-\cos(6x)}{x}\).

Evaluate \(\lim_{x\to0}\frac{\sin(2x)+\sin(5x)}{x}\).

Evaluate \(\lim_{x\to0}\frac{\sin(4x)}{\tan(9x)}\).

Evaluate \(\lim_{x\to0}\frac{\sin(3x)-\sin x}{x}\).

A small-angle pendulum model uses \(\sin\theta\approx\theta\). Explain the limit statement that justifies replacing \(\sin\theta/\theta\) by \(1\) as \(\theta\to0\).

Evaluate \(\lim_{x\to0}\frac{\sin(2x)\sin(5x)}{x^2}\).

Evaluate \(\lim_{x\to0}\frac{\tan(3x)-\sin(3x)}{x}\).

Find \(k\) if \(\lim_{x\to0}\frac{\sin(kx)}{x}=12\), where \(k\) is a constant.

Find all constants \(a\) for which \(\lim_{x\to0}\frac{\sin(ax)}{\sin(2x)}=3\).

For constants \(a\) and \(b\ne0\), determine \(\lim_{x\to0}\frac{\sin(ax)}{\tan(bx)}\).

A student claims \(\lim_{x\to0}\frac{\sin(5x)}{2x}=1\) because \(\sin u/u\to1\). Identify the error and give the correct limit.

Use squeeze-theorem reasoning to justify why \(\lim_{x\to0^+}\frac{\sin x}{x}=1\).

Explain why the statement \(\lim_{x\to0}\frac{\sin x}{x}=1\) would not be correct if \(x\) were measured in degrees inside the sine function.