Answer
$$
f(x)=\frac{5}{x}+\frac{2 x}{x+4}
$$
is continuous for all real numbers $x$ except $x = 0 $, $x = -4 $.
Work Step by Step
$$
f(x)=\frac{5}{x}+\frac{2 x}{x+4}
$$
suppose
$$
f(x)=g(x)+h(x)
$$
such that
$$g(x)= \frac{5}{x} , \quad \quad h(x)=\frac{2 x}{x+4} $$
Solving the equation
$$
x+4=0
$$
yields discontinuities at $x = -4$.
Therefore the function
$$
h(x)=\frac{2 x}{x+4}
$$
is continuous for all real number $x$ except $ x = -4$
Thus
$$
f(x)=\frac{5}{x}+\frac{2 x}{x+4}
$$
is continuous for all real numbers $x$ except $x = 0 $, $x = -4 $.
Finally,
values of $x$, at which $f $ is not continuous are $x = 0$, $x = -4$.
