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.

Explore the left-hand limit of -1/x as x approaches zero, analyzing how -1/x diverges towards positive infinity. This guide includes step-by-step solutions and concepts in limits, infinity, and one-sided limits. Ideal for students in Grades 11-12 studying calculus.

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