Question

Find the limit. use l'hospital's rule where appropriate. if there is a more elementary method, consider using it. lim x→0 sin(9x) csc(7x)

Answer 1

$\displaystyle\lim_{x\to0}\sin9x\csc7x=\lim_{x\to0}\frac{\sin9x}{\sin7x}=\frac97\lim_{x\to0}\frac{7x\sin9x}{9x\sin7x}$

Recall that for $a\neq0$,

$\displaystyle\lim_{x\to0}\frac{\sin ax}{ax}=\lim_{x\to0}\frac{ax}{\sin ax}=1$

You can separate the two expressions with $a=7$ and $a=9$, which both approach 1, so that the limit is $\dfrac97$.

Alternatively, using L'Hopital's rule,

$\displaystyle\lim_{x\to0}\frac{\sin9x}{\sin7x}\stackrel{\mathrm{LHR}}=\lim_{x\to0}\frac{9\cos9x}{7\cos7x}=\dfrac97$