To find the limit of $\lim_{x \to 0^{-}} -\frac{1}{x}$, we need to analyze the behavior of the function $-\frac{1}{x}$ as $x$ approaches 0 from the left (i.e., $x$ is negative and approaches zero).

Step-by-Step Solution:

1. Expression Analysis: The function is given as $-\frac{1}{x}$.

2. Behavior Near Zero from the Left:

   • As $x$ approaches 0 from the left (negative values), $x$ is a small negative number.
   • The reciprocal $\frac{1}{x}$ will be a large negative number because dividing 1 by a small negative number results in a large negative magnitude.
   • Since there is a negative sign in front, $-\frac{1}{x}$ will become a large positive number.

3. Conclusion: Therefore, $\lim_{x \to 0^{-}} -\frac{1}{x} = +\infty$.

The limit diverges to positive infinity as $x$ approaches zero from the left.

Would you like further details, or do you have any other questions?

Related Questions:

1. What is $\lim_{x \to 0^{+}} -\frac{1}{x}$?
2. How does the behavior of $\frac{1}{x}$ differ from $-\frac{1}{x}$ as $x$ approaches zero?
3. What is the limit of $-\frac{1}{x^2}$ as $x$ approaches zero from the left?
4. How does the function $-\frac{1}{x}$ behave as $x$ approaches infinity?
5. How do we find the limit of a function at infinity?

Tip:

When finding limits as $x$ approaches zero, always check if the approach is from the left ($x \to 0^-$) or from the right ($x \to 0^+$), as the behavior may differ significantly.