Compute the limits.
\begin{alignat*}{3}
&a) \lim_{x \to 0} {\frac{\sin^2{2x}}{ x^2}} & &b) \lim_{x \to 0}{\frac{1-\cos{3x} }{ x^2}}
& &c) \lim_{x \to 0}{\frac{x}{ 1-\sec x}} \\
&d) \lim_{x \to 0}{\frac{1+\cot^2 {3x} }{ 4x^2}} & &e) \lim_{x\to \frac{\pi }{ 6}}
{[ 1-\sec {(x+\frac{\pi }{ 3})}] (x - \frac{\pi }{ 6})^2} & &f)
\lim_{x\to \frac{\pi }{ 2}}{\frac{\cos x }{ 2x - \pi}}
\end{alignat*}
Tags: trig-limits, limits, trig-identities, trig, tricks, computation, a--a, t2, c1
This is what can only be described as a set of trig tricks. (a) is a disguised use of the trig limit “sin x / x”, (c) can be massaged into a “1 - cos x/x” form, as can (f) after a substitution x → x - Pi/2. (b) is easiest with multiplication of the top and bottom by the conjugate 1 + cos 3x, though you can also see it if you know the power series expansion of cosine. This is unlikely for 1st semester calculus. (d) can be seen to be diverging, and (e) simplifies immensely by substituting t = x - Pi/6 and recalling that cos(t+Pi/2)=-sin(t). This would be a III technical difficulty for any class that hasn’t covered these tricks.
– Eric Hsu 1998-08-09