[tex]\displaystyle\lim_{x\to0}\sin9x\csc7x=\lim_{x\to0}\frac{\sin9x}{\sin7x}=\frac97\lim_{x\to0}\frac{7x\sin9x}{9x\sin7x}[/tex]
Recall that for [tex]a\neq0[/tex],
[tex]\displaystyle\lim_{x\to0}\frac{\sin ax}{ax}=\lim_{x\to0}\frac{ax}{\sin ax}=1[/tex]
You can separate the two expressions with [tex]a=7[/tex] and [tex]a=9[/tex], which both approach 1, so that the limit is [tex]\dfrac97[/tex].
Alternatively, using L'Hopital's rule,
[tex]\displaystyle\lim_{x\to0}\frac{\sin9x}{\sin7x}\stackrel{\mathrm{LHR}}=\lim_{x\to0}\frac{9\cos9x}{7\cos7x}=\dfrac97[/tex]
