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Understanding the concept of limits and continuity is fundamental in calculus. It involves analyzing the behavior of a function as the input (or variable) approaches a certain value. Imagine you're on a path that's getting closer and closer to a point, but you're instructed not to step on it. The value that you are approaching is the limit.

Continuity, on the other hand, implies that as you walk along this path, you can do so without any jumps or gaps. For a function to be continuous at a point, it must meet three conditions: it has to be defined at that point, the limit must exist as it approaches that point, and the limit must equal the function's value at that point. This trio ensures a smooth, uninterrupted path in the function's graph.

In our exercise, we are seeing how the function \( \sqrt{2x^3} - \sqrt{2}x \) behaves as \( x \) approaches 2. We are not concerned with what happens at 2, just as it gets close. This is the essence of studying limits. The continuity would be checked by evaluating the function at \( x = 2 \) and confirming it matches the limit we calculated.

Limit laws are a set of tools that make finding limits easier. They're like shortcuts that help you understand and solve limits without the heavy lifting of complex calculations every single time. These laws allow you to break down complicated expressions into simpler parts, work with them individually, and then recombine them.

Some of the most basic limit laws include the sum law, the product law, and the quotient law, which allow you to take the limit of sums, products, and quotients, respectively, term by term. There’s also the power law that comes in handy when dealing with exponents. In our step by step solution, we implicitly used these laws when we simplified the expression by factoring out \( \sqrt{2}x \) and then finding the limit of each part separately.

\textbf{Example of Limit Laws}:
Given the limit law for products, we can say \( \lim_{x \rightarrow a} [f(x)g(x)] = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x) \), provided these individual limits exist. This principle helps in breaking down and assessing the behavior of more intricate functions.

Square root functions are a type of radical function where the main operation is taking a square root. The general form is \( f(x) = \sqrt{x} \), where \( x \) must be a non-negative number since we can't take the square root of a negative number in the set of real numbers.

Square root functions are interesting because they involve half exponents, \( x^{1/2} \) and can introduce non-linear behavior in what would otherwise be linear expressions. They have a distinctive graph shape, resembling the right half of a sideways parabola. In the context of limits, square root functions can sometimes complicate matters due to their undefined nature for negative inputs. However, they follow the same limit laws when combined with other algebraic operations.

When looking at the limit of a square root function like in our exercise, factoring can simplify the process—just as we extracted \( \sqrt{2}x \) from \( \sqrt{2x^3} \)—making it easier to pinpoint the limit as \( x \) approaches our value of interest.