### 5.24. Investigate the function

$$
f(x)=\frac{5^{1 / x}}{1+5^{1 / x}}
$$

for continuity and determine the nature of the points of discontinuity.
Solution. The function is defined everywhere except at the point $x=0$. Let's investigate the behavior of the function in the neighborhood of the point $x=0$. We will find the one-sided limits

$$
\lim _{x \rightarrow 0-0} f(x)=\lim _{x \rightarrow 0-0} \frac{5^{1 / x}}{1+5^{1 / x}}=0
$$

since as $x \rightarrow 0-0$, i.e., from the left of zero, $1 / x \rightarrow-\infty \Rightarrow$ $5^{1 / x} \rightarrow 0$.

$$
\lim _{x \rightarrow 0+0} f(x)=\lim _{x \rightarrow 0+0} \frac{5^{1 / x}}{1+5^{1 / x}}=\lim _{x \rightarrow 0+0} \frac{1}{5^{-1 / x}+1}=1
$$

since as $x \rightarrow 0+0$, $1 / x \rightarrow+\infty \Rightarrow 5^{-1 / x} \rightarrow 0$.

Thus, the one-sided limits of the function at the point $x=0$ exist but are not equal to each other. Therefore, the function $f(x)$ has a discontinuity of the first kind at the point $x=0$.