To find the limit of the function \( (x-1)^2 \) as \( x \) approaches 1, we can evaluate it directly since the function is continuous around the point \( x = 1 \).

\[
\lim_{x \to 1} (x-1)^2
\]

### Step-by-Step Solution:

1. **Substitute \( x = 1 \) into the function:**

   The expression \( (x-1)^2 \) represents a polynomial function, which is continuous everywhere. Therefore, we can directly substitute \( x = 1 \) into the expression.

2. **Calculate the expression:**

   \[
   (x-1)^2 = (1-1)^2 = 0^2 = 0
   \]

### Conclusion:

\[
\lim_{x \to 1} (x-1)^2 = 0
\]

Would you like more details or have any further questions?

### Additional Questions:

1. How do you find the limit of \( (x-1)^n \) as \( x \) approaches 1 for different values of \( n \)?
2. What is the general approach to finding the limits of polynomial functions?
3. How can you use the limit definition to prove the result formally?
4. What happens to the limit if the power in the function changes, such as \( (x-1)^3 \)?
5. How does the continuity of a function affect its limit behavior?

### Tip:

For continuous functions, you can often evaluate limits directly by substitution, making these problems straightforward to solve.

Learn how to evaluate the limit of the function (x-1)^2 as x approaches 1 with a detailed step-by-step solution. This problem covers key mathematical concepts including limits, continuity, and polynomial functions. Suitable for students in Grades 10-12.

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