limit of function

Question : Evaluate the following limit \[\lim_{x\rightarrow\frac{\pi}{2}\ }{\frac{\sin{\left(2x\right)}}{x^2+\frac{\pi}{2}x-\frac{\pi^2}{2}}}\] Solution : To evaluate the limit \[\lim_{x\rightarrow\frac{\pi}{2}\ }{\frac{\sin{\left(2x\right)}}{x^2+\frac{\pi}{2}x-\frac{\pi^2}{2}}}\] Since \[sin(2\ast\frac{\pi}{2})=sin(\pi)=0 \] and \[\left(\frac{\pi}{2}\right)^2+\frac{\pi}{2}\ast\frac{\pi}{2}-\frac{\pi^2}{2}=\frac{\pi^2}{4}+\frac{\pi^2}{4}-\frac{\pi^2}{2}=\frac{\pi^2}{2}-\frac{\pi^2}{2}=0\] That is \[\lim_{x\rightarrow\frac{\pi}{2}\ }{\frac{\sin{\left(2x\right)}}{x^2+\frac{\pi}{2}x-\frac{\pi^2}{2}}} \] Is in \[\frac{0}{0}\ form\]…