Section 1.3 #71: Compute $\displaystyle\lim_{x \rightarrow \frac{\pi}{2}} \dfrac{\cos(x)}{\cot(x)}.$
Solution: Directly substituting $\dfrac{\pi}{2}$ into $\dfrac{\cos(x)}{\cot(x)}$ yields indeterminant form $\dfrac{0}{0}$, so there is more work to do. Recall that $\cot(x) = \dfrac{\cos(x)}{\sin(x)}$. Algebra shows us that $$\dfrac{\cos(x)}{\cot(x)} = \dfrac{\cos(x)}{\frac{\cos(x)}{\sin(x)}} = \cos(x) \dfrac{\sin(x)}{\cos(x)}=\sin(x).$$ Therefore we compute $$\displaystyle\lim_{x \rightarrow \frac{\pi}{2}} \dfrac{\cos(x)}{\cot(x)} = \displaystyle\lim_{x \rightarrow \frac{\pi}{2}} \sin(x) = \sin \left( \dfrac{\pi}{2} \right) = 1.$$
Section 1.4, #22: Compute the limit or explain why it does not exist: $\displaystyle\lim_{x \rightarrow \frac{\pi}{2}} \sec(x)$.
Solution: Recall that $\sec(x)$ has an asymptote at $x=\dfrac{\pi}{2}$. Therefore the limit does not exist.
Section 1.5, #27: Find the vertical asymptotes of the function $s(t)=\dfrac{t}{\sin(t)}$.
Solution: We find vertical asymptotes by setting the denominator equal to zero. Solving the equation $\sin(x)=0$ yields the solution $t=k \pi$ for any (nonzero) integer $k$. Therefore the function $s$ has asymptotes at $t=k \pi$ for any integer $k$.
note: we require nonzero here because we know that $\displaystyle\lim_{x \rightarrow 0} s(t)=1$ by Theorem 1.9 on page 65, which indicates that we have no asymptote since asymptotes "blow up" to $\infty$ or $-\infty$