5.25. Investigate the function

$$
f(x)=\frac{\sin 3 x}{x}
$$

for continuity and determine the nature of the points of discontinuity.
Solution. The one-sided limits of this function exist and are equal to 3:

$$
\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}=3 \lim _{x \rightarrow 0} \frac{\sin t}{t}=3
$$

(a substitution $3 x=t$ has been made).

At the point $x=0$, the function is undefined. Thus, at the point $x=0$, we have a removable discontinuity. This discontinuity is removed by defining the function to have the value 3 at the point $x=0$. The new function

$$
g(x)=\left\{\begin{array}{l}
\frac{\sin 3 x}{x}, x \neq 0 \\
3, x=0
\end{array}\right.
$$

will be continuous at the point $x=0$.