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.