Question

lf the function $f\left(x\right)=\{\begin{array}{c}\frac{{\sin }^{2}ax}{x^2},x\ne 0\\ 1,x=0\end{array}$ is continuous at $x=0$ then $a=$

• A. $\pm 1$
• B. $0$
• C. $\pm \frac{1}{2}$
• D. $\pm \frac{1}{3}$

Answer 1

Answer: (A)

$\pm 1$

Since, the given function,
$f\left(x\right)=\frac{{\sin }^{2}ax}{x^2}$ is continuous at $x=0$, then
$f\left(0\right)=\underset{x\to 0}{lim}\frac{{sin}^{2}ax}{x^2}$
$\underset{x\to 0}{lim}\frac{{sin}^{2}ax}{x^2}=\underset{x\to 0}{lim}\frac{{\left(\sin \left(ax\right)\right)}^2}{{\left(ax\right)}^2}\times a^2$
$=1\times a^2$
Since, the function is continuous at x = 0
$a^2=1$
Or, $a=1,-1$