## Explanation

### Understanding the Limit

When dealing with the limit of the function $\frac{1}{x}$ as $x$ approaches 0, it's crucial to analyze the behavior of the function from both the left and right sides of 0.

### Left-hand limit $(x \to 0^{-})$

When $x$ approaches 0 from the left, or from negative values, the denominator $x$ is negative but getting very small in magnitude:

$\lim_{x \to 0^{-}} \frac{1}{x}$As $x$ gets closer to 0 from the negative side, $\frac{1}{x}$ becomes a very large **negative** number. Thus:

### Right-hand limit $(x \to 0^{+})$

Similarly, when $x$ approaches 0 from the right, or from positive values, the denominator $x$ is positive but also getting very small in magnitude:

$\lim_{x \to 0^{+}} \frac{1}{x}$As $x$ gets closer to 0 from the positive side, $\frac{1}{x}$ becomes a very large **positive** number. Hence:

### Conclusion

Since the left-hand limit and the right-hand limit are not equal:

$\lim_{x \to 0^{-}} \frac{1}{x} \neq \lim_{x \to 0^{+}} \frac{1}{x}$we conclude that:

$\mathbf{\lim_{x \to 0} \frac{1}{x} \text{ does not exist}}$This discrepancy between the left-hand and right-hand limits means that the overall limit of $\frac{1}{x}$ as $x$ approaches 0 **does not exist**.