limit of( x cube -1)/x square when x approaches zero.solve using limit theorem and methods by algebraically for existence.

We are tasked with evaluating the limit:

\[
\lim_{x \to 0} \frac{x^3 - 1}{x^2}
\]

Let's solve this step by step using algebraic methods and limit theorems.

### Step 1: Direct Substitution
First, attempt direct substitution by plugging \(x = 0\) into the expression.

\[
\frac{0^3 - 1}{0^2} = \frac{-1}{0}
\]

Since division by zero is undefined, we cannot directly substitute \(x = 0\). This suggests that we need a different approach, such as simplifying the expression.

### Step 2: Rewrite the Expression
We can simplify the expression by separating the terms of the fraction:

\[
\frac{x^3 - 1}{x^2} = \frac{x^3}{x^2} - \frac{1}{x^2}
\]

This simplifies further to:

\[
x - \frac{1}{x^2}
\]

### Step 3: Analyze the Behavior as \(x \to 0\)
Now, evaluate the limit for each term separately.

- The first term, \(x\), clearly approaches 0 as \(x \to 0\).
- The second term, \(-\frac{1}{x^2}\), approaches \(-\infty\) as \(x \to 0\) because \(\frac{1}{x^2}\) grows without bound when \(x\) is very small and close to 0.

Thus, the entire expression:

\[
x - \frac{1}{x^2}
\]

will approach \(-\infty\) as \(x \to 0\).

### Step 4: Conclusion
Therefore, the limit does not exist in the usual sense because it tends to \(-\infty\), which indicates that the function diverges as \(x\) approaches 0.

\[
\boxed{\lim_{x \to 0} \frac{x^3 - 1}{x^2} = -\infty}
\]

### Would you like to go into more details, or have any questions? 

Here are 5 related questions for further exploration:
1. What is the limit of \(\frac{x^3 - 1}{x^2}\) as \(x\) approaches infinity?
2. How do limits involving infinity differ from finite limits?
3. What is the behavior of \( \frac{x^n - 1}{x^m} \) for general \(n\) and \(m\) as \(x\) approaches zero?
4. How does one handle indeterminate forms like \( \frac{0}{0} \) using L'Hopital's Rule?
5. How can we graphically interpret the behavior of this function near zero?

**Tip**: When encountering limits that tend toward infinity or indeterminate forms, simplifying the function or using tools like L'Hopital's Rule can often clarify the behavior of the function.

limit of (x^3 - 1) / x^2 when x approaches zero. Solve using limit theorem and methods algebraically for existence.

Explore how to solve the limit of (x^3 - 1) / x^2 as x approaches zero using algebraic simplification and limit theorems. Understand the behavior of the function, how division by zero impacts the solution, and why the limit tends to negative infinity. This detailed solution is ideal for Grades 11-12 and undergraduate students.

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