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The given expression is:

\[
\lim_{{h \to 0}} \left( (x + h)^2 - x^2 \right)
\]

We need to evaluate this limit while treating \(x\) as a constant.

### Step 1: Expand the squared term
First, expand \((x + h)^2\):

\[
(x + h)^2 = x^2 + 2xh + h^2
\]

### Step 2: Substitute into the limit expression
Now substitute this expanded form into the limit:

\[
\lim_{{h \to 0}} \left( x^2 + 2xh + h^2 - x^2 \right)
\]

Simplifying the expression:

\[
= \lim_{{h \to 0}} \left( 2xh + h^2 \right)
\]

### Step 3: Factor and evaluate the limit
We can factor out \(h\):

\[
= \lim_{{h \to 0}} h(2x + h)
\]

As \(h \to 0\), the expression \(h(2x + h)\) tends to 0, because both \(h\) and \(h^2\) vanish.

Thus, the limit is:

\[
\boxed{0}
\]

Would you like more details, or do you have any other questions?

Here are five related questions you might find useful:
1. How do we apply the power rule for limits when dealing with higher-order terms?
2. What is the formal definition of a limit, and how does it apply to this problem?
3. How can this limit problem be related to finding the derivative of \(x^2\)?
4. What happens if we don't treat \(x\) as a constant in the limit process?
5. Can this limit be evaluated using L'Hopital's Rule, and if so, how?

**Tip:** Understanding the expansion of algebraic expressions helps simplify limits and derivatives in calculus.

Learn how to calculate the limit of the expression (x+h)^2 - x^2 as h approaches 0, treating x as a constant. This problem involves key calculus and algebra concepts, such as limits and binomial expansion, and is suitable for students in Grades 10-12.

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