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Imagine you're on a journey towards a fixed destination, but with each step, you halve the distance between yourself and the goal. Theoretically, you are ever closer but never quite there, yet for practical purposes, we can say you'll eventually arrive. A similar idea applies to a convergent sequence in mathematics.

When a sequence's terms get closer and closer to a specific value as you move through the sequence, it's called a convergent sequence. The fixed destination is akin to the sequence's limit. For instance, the given problem involves the sequence \(\sqrt{n^{2}+n}-n\). By applying algebraic techniques and understanding the behavior of the sequence as \(n\) grows larger, we determine that the destination, or the limit, is \(\frac{1}{2}\). Convergence essentially tells us where the sequence's journey ends after an infinite number of steps.

The concept of infinite limits guides us through scenarios where our function's values either increase without bound or decrease without bound as we approach a certain point. This is distinctly different from what we see in convergent sequences, which approach a specific number. We're not talking about the limit being infinitely large, but rather considering what happens as the variable, in this case \(n\), heads towards infinity.

In the given sequence, we explore the limit as \(n\) grows indefinitely - a journey into the infinite. But instead of the values blowing up to infinity, they settle down, approaching the finite value of \(\frac{1}{2}\). This is a beautiful illustration that even as we venture towards infinity with our variable, the sequence itself may still comfortably converge to a finite limit.

The difference of squares is a nifty algebraic pattern that surfaces quite frequently in mathematical problems. It is when you have two terms squared and subtracted from one another, represented as \(a^2 - b^2\). This special form allows us to factor the expression into \(a + b)(a - b)\), which is exactly what's employed in the solution to our original exercise.

By multiplying the sequence's terms \(\sqrt{n(n+1)} - n\) by its conjugate \(\sqrt{n(n+1)} + n\), we use the difference of squares idea to simplify the expression. The result? A cleaner and more manageable form that ultimately leads us to the convergence point. Spotting a difference of squares in the wild can be a crucial step to unraveling more complex problems, just like this one!

The limit of a function is a foundational concept within calculus that describes the behavior of a function as the input approaches a particular value. It doesn't have to be about what the function equals at that point, but where it's 'headed'.

In our exercise, we investigate the limit as \(n\) approaches infinity, which tells us about the long-term behavior of the sequence \(\sqrt{n^{2}+n}-n\). The limit concept is seamlessly woven into the exercise, as we break down and simplify the expression, and analyze how the terms behave in the grand scale of growing numbers. This grasp on the concept of limits is essential not just for single problems, but for an entire array of calculus-related topics.