We want to evaluate the limit:

$\lim_{x \to 0^+} \left( \frac{x^2}{2} - \frac{1}{x} \right).$

Step-by-Step Solution:

1. Split the expression: The given function is: $f(x) = \frac{x^2}{2} - \frac{1}{x}.$ As $x \to 0^+$, examine the behavior of each term separately.

2. Behavior of $\frac{x^2}{2}$:

   • When $x \to 0^+$, $x^2 \to 0$, so $\frac{x^2}{2} \to 0$.

3. Behavior of $-\frac{1}{x}$:

   • As $x \to 0^+$, $1/x \to +\infty$, so $-1/x \to -\infty$.

4. Combine the terms:

   • The term $-\frac{1}{x}$ dominates because it approaches $-\infty$ much faster than $\frac{x^2}{2}$ approaches $0$.

5. Conclusion: $\lim_{x \to 0^+} \left( \frac{x^2}{2} - \frac{1}{x} \right) = -\infty.$

Final Answer:

$\boxed{-\infty}$

---

Do you want further details on the steps or behavior of similar limits? Let me know! 😊

---

Related Questions:

1. What happens to the limit if the term $-1/x$ is replaced with $+1/x$?
2. How does the behavior of $\frac{x^2}{2}$ change if $x$ approaches a nonzero value?
3. What if the term $-1/x$ is replaced with $-1/x^2$?
4. Can the expression $\frac{x^2}{2} - \frac{1}{x}$ ever cross zero for $x > 0$?
5. What is the behavior of this function as $x \to \infty$?

Tip:

When evaluating limits involving dominant terms, focus on the one with the fastest growth or decay near the point of interest.