Question Number 73569 by Rio Michael last updated on 13/Nov/19
![let f(x) = ((tanx)/(tan2x)) . Find the points of discontinuity of f on [0,2π] and determine wether each duscontinuity is a point discontinuity,a jump discontinuity,or a vertical asymtote](Q73569.png)
$${let}\:{f}\left({x}\right)\:=\:\frac{{tanx}}{{tan}\mathrm{2}{x}}\:.\:{Find}\:{the}\:{points}\:{of}\:{discontinuity} \\ $$$${of}\:{f}\:{on}\:\left[\mathrm{0},\mathrm{2}\pi\right]\:{and}\:{determine}\:{wether}\:{each}\:{duscontinuity}\:{is} \\ $$$${a}\:{point}\:{discontinuity},{a}\:{jump}\:{discontinuity},{or}\:{a}\:{vertical}\:{asymtote} \\ $$$$ \\ $$
Commented by Rio Michael last updated on 13/Nov/19
$${i}\:{appreciate} \\ $$
Commented by kaivan.ahmadi last updated on 13/Nov/19
$${f}\left({x}\right)=\frac{\frac{{sinx}}{{cosx}}}{\frac{{sin}\mathrm{2}{x}}{{cos}\mathrm{2}{x}}}=\frac{{sinxcos}\mathrm{2}{x}}{{cosxsin}\mathrm{2}{x}} \\ $$$${cosx}=\mathrm{0}\Rightarrow{x}=\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$$${sin}\mathrm{2}{x}=\mathrm{0}\Rightarrow{x}=\mathrm{0},\frac{\pi}{\mathrm{2}},\pi \\ $$$$\Rightarrow{f}\:{is}\:{not}\:{continuous}\:{at}\:\left\{\mathrm{0},\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}},\pi\right\}. \\ $$