\[y=\frac{tanx}{sin(2x)+1}\]

.We can some asymptotes by setting the denominator equal to zero.

\[sin(2x)+1=0 \rightarrow sin(2x)=-1 \rightarrow 2x=(2n+ \frac{3}{2}) \pi \rightarrow x=(n+ \frac{3}{4}) \pi\]

We cannot make

\[x\]

the subject, so cannot perform the same analysis to find asymptotes for \[y\]

. The function \[\frac{tanx}{sin(2x)+1}\]

contains only trigonometric functions - no terms or factors such as \[e^x, \: x^3\]

for example. This means that the function is periodic, in this case the function repeats every \[\pi\]

radians. Because of this there can be no \[x\]

asymptotes, or any other asymptotes of any kind. There only asymptotes are parallel to the \[y\]

axis, given above.