State the leading behaviour of \(\sin x\) as \(x\) approaches \(0\).

State the leading behaviour of \(1-\cos x\) as \(x\) approaches \(0\).

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

Evaluate \(\lim_{x\to0}\frac{e^x-1}{x}\).

Evaluate \(\lim_{x\to0}\frac{1-\cos x}{x^2}\).

Evaluate \(\lim_{x\to0}\frac{e^x-1-x}{x^2}\).

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

Evaluate \(\lim_{x\to0}\frac{\ln(1+x)}{x}\).

Evaluate \(\lim_{x\to0}\frac{x-\arctan x}{x^3}\), using \(\arctan x=x-\frac{x^3}{3}+\cdots\).

Evaluate \(\lim_{x\to0}\frac{e^{3x}-1-3x}{x^2}\).

Why may a Taylor-limit calculation require more than the first term?

Why do trigonometric Taylor limits assume radians?

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

Evaluate \(\lim_{x\to0}\frac{\cos x-1+\frac{x^2}{2}}{x^4}\).

Evaluate \(\lim_{x\to0}\frac{\ln(1+x)-x+\frac{x^2}{2}}{x^3}\).

Evaluate \(\lim_{x\to0}\frac{e^x+e^{-x}-2}{x^2}\).

Evaluate \(\lim_{x\to0}\frac{\sin(2x)-2\sin x}{x^3}\).

Diagnose the error: \(\frac{1-\cos x}{x}\) tends to \(\frac12\) because \(1-\cos x\) behaves like \(\frac{x^2}{2}\).

How do you handle a Taylor-limit quotient when leading terms cancel?

Why do \(\sin\theta\approx\theta\) and \(1-\cos\theta\approx\frac{\theta^2}{2}\) have different orders?