Question

Show that the function $f\left(x\right)=\{\begin{array}{c}\frac{{\sin }^{2}ax}{x^2},whenx\ne 0\\ a^2,whenx=0.\end{array}$ is discontinuous at $x=0$

Redefine the function in such a way that it becomes continuous at $x=a$.

Answer 1

Clearly,

$\begin{array}{c}f\left(0\right)=1\\ Also,\\ {lim}_{x\to 0}f\left(x\right)={lim}_{x\to 0}\frac{{\sin }^{2}ax}{x^2}=a^2.{lim}_{ax\to 0}{\left(\frac{\sin ax}{ax}\right)}^2=a^2.\\ Since,\\ {lim}_{x\to 0}f\left(x\right)\ne f\left(0\right),\\ f\left(x\right)isdiscontinuousatx=0\\ However,itbecomescontinuousatf\left(0\right)=a^2\\ So,thedesiredfunctionisf\left(x\right)=\{\begin{array}{c}\frac{{\sin }^{2}ax}{x^2},whenx\ne 0\\ a^2,whenx=0.\end{array}\end{array}$