Question

$f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{\sin 2x}{x}& x\ne 0\\ 1& x=0\end{array}\end{array}$

Check whether $f\left(x\right)$ has continuity at $x=0$

Answer 1

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{\sin 2x}{x}$

$=\underset{x\to 0^{+}}{lim}\frac{\sin 2x}{2x}2=2$

At $x=0$ $f\left(x\right)=1$

$\therefore$ $\underset{x\to 0^{+}}{lim}f(x)\ne f\left(0\right)$

Therefore, the function is not continuous at $x=0$