Example 7. Find the one-sided limits of the function $f(x)=3^{\frac{1}{x}}$ as $x \rightarrow 0$ from the left and right.
Solution.

Let $x$ tend to 0, remaining less than it, i.e., $x \rightarrow -0$, then $x$ will be an infinitely small quantity, taking negative values, and the quantity $\frac{1}{x}$ will be a negative infinitely large quantity (see formula (1.41)). This function can be represented as follows:

$$
3^{\frac{1}{x}} = 3^{-\frac{1}{|x|}} = \frac{1}{3^{\left|\frac{1}{x}\right|}}
$$

Since $\frac{1}{|x|}$ is infinitely large as $x \rightarrow 0$, then $3^{\frac{1}{|x|}}$ is also infinitely large, and its reciprocal, i.e., $\frac{1}{3^{\left|\frac{1}{x}\right|}}$, will be infinitely small (see formula (1.40)). Therefore,

$$
\lim _{x \rightarrow -0} 3^{\frac{1}{x}} = \lim _{x \rightarrow -0} 3^{-\frac{1}{|x|}} = \lim _{x \rightarrow -0} \frac{1}{3^{\left|\frac{1}{x}\right|}} = 0
$$

Let $x$ tend to 0, remaining greater than it, i.e., $x \rightarrow +0$, then $x$ will be a positive infinitely small quantity, and $\frac{1}{x}$ will be a positive infinitely large quantity, and the quantity $3^{\frac{1}{x}}$ will also be a positive infinitely large quantity, i.e.,

$$
\lim _{x \rightarrow +0} 3^{\frac{1}{x}} = \lim _{x \rightarrow +0} 3^{\frac{1}{|x|}} = +\infty
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-037.jpg?height=402&width=506&top_left_y=82&top_left_x=55)

Figure 1.23
Since the limits are not equal to each other, the limit of the function as $x \rightarrow 0$ in any arbitrary manner does not exist.

The graph of the function $f(x) = 3^{\frac{1}{x}}$ is shown in Figure 1.23.