Question

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

Answer 1

The function $f\left(x\right)=\left\{\begin{array}{cc}\frac{{\sin }^{2}ax}{x^2},& x\ne 0\\ 1,& x=0\end{array}\right.$ is continuous at x = 0.

$\therefore \underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

$\Rightarrow \underset{x\to 0}{\lim }\frac{{\sin }^{2}ax}{x^2}=1$

$\Rightarrow a^2{\left(\underset{x\to 0}{\lim }\frac{\sin ax}{ax}\right)}^2=1$

$\Rightarrow a^2\times {\left(1\right)}^2=1$ $\left(\underset{x\to 0}{\lim }\frac{\sin x}{x}=1\right)$

$\Rightarrow a^2=1$

$\Rightarrow a=\pm 1$

Thus, the value of a is ±1.


If the function $f\left(x\right)=\left\{\begin{array}{cc}\frac{{\sin }^{2}ax}{x^2},& x\ne 0\\ 1,& x=0\end{array}\right.$is continuous at x = 0, then a = ___±1___.