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Understanding limits is essential in calculus as it deals with the behavior of functions as they approach a specific point. The limit of a function at a certain point describes what value the function is approaching as its input gets infinitely close to that point. In the provided exercise, the limit is used to define the behavior of the function as the variable x grows larger, specifically \( \lim _{x \rightarrow \infty} f(x)=2 \). This indicates that as x increases without bound, the function approaches the value of 2. This concept is closely related to continuity, which is a condition where a function has no breaks, jumps, or holes at or around a given point. If a function is continuous at a point, then the limit of the function as it approaches that point is equal to the function's value at that point. In our exercise, although continuity isn't explicitly stated, it's implied in how the function behaves around x=1.

Derivatives are fundamental in calculus as they represent the rate of change of a function. They give us information about the slope of the function's graph at any given point. When the derivative, denoted as \(f'(x)\), is positive, the function is increasing. Conversely, if the derivative is negative, the function is decreasing. In the exercise, \(f'(x) < 0\) for \(x<1\), indicating that the graph is going downwards as it approaches the point (1,0) from the left. On the other end, \(f'(x) > 0\) for \(x>1\), suggesting the graph is ascending after the point (1,0). This behavior at x=1, where the derivative equals zero, \(f'(1)=0\), suggests a turning point in the graph. By calculating and interpreting derivatives, we obtain valuable insights into the graph's shape and behavior at different intervals.

Horizontal asymptotes are horizontal lines that the graph of a function approaches as its inputs either increase or decrease without bound. They provide a way to understand the long-term behavior of a function's graph. In simple terms, as the value of x becomes very large or very small, the value of the function will get closer and closer to the horizontal asymptote but never actually reach it. In the current exercise, the horizontal asymptote is drawn at \(y = 2\), based on the limit \(\lim _{x \rightarrow \infty} f(x) = 2\). This means that as x increases to infinity, the function's value will approach and hover near y=2, but it won't cross this horizontal barrier. Graphing horizontal asymptotes is a critical step for understanding the end behavior of a function and is an essential aspect of graphing functions efficiently and accurately.