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The given limit problem is:

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

Let's simplify and solve it step by step.

### Step 1: Simplify the expression inside the square root

The expression under the square root is \( 1 + 2x - x - 1 \), which simplifies to:
\[
1 + 2x - x - 1 = x
\]

Thus, the limit becomes:
\[
\lim_{x \to 0} \frac{x^2}{\sqrt{x}} = \lim_{x \to 0} \frac{x^2}{x^{1/2}}
\]

### Step 2: Simplify the fraction

We can simplify the expression:
\[
\frac{x^2}{x^{1/2}} = x^{2 - 1/2} = x^{3/2}
\]

### Step 3: Evaluate the limit

Now, we take the limit as \(x \to 0\):
\[
\lim_{x \to 0} x^{3/2} = 0
\]

Thus, the solution to the limit is:
\[
\boxed{0}
\]

Would you like further clarification or details on any of the steps?

Here are 5 related questions you might find interesting:

1. What is the difference between the exponent rules when dividing powers of \(x\)?
2. How can L'Hopital's Rule be applied to solve indeterminate forms of limits?
3. How do you evaluate limits that involve square roots or other radicals?
4. How do you identify and resolve indeterminate forms like \(\frac{0}{0}\)?
5. How can limits help in understanding the behavior of functions near points of interest?

**Tip**: Always simplify expressions as much as possible before attempting to evaluate limits. This helps to identify the limit behavior clearly.

lim_{x → 0} (x^2 / sqrt(1 + 2x - x - 1))

This problem explores the limit of a rational function involving square roots as x approaches 0. The solution simplifies the expression and applies basic limit evaluation rules, suitable for students in Grades 11-12.

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