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One of the fundamental concepts in calculus is the continuity of a function. Continuous functions are those that do not have any abrupt changes or holes in their graphs. In simpler terms, you can draw the graph of a continuous function without lifting your pen from the paper.

When dealing with limits and continuous functions, an important aspect is that they promise predictability around a point. In the case of continuous functions, the limit as you approach any point 'a' is simply the value of the function at 'a'. That is, \(lim_{{x \rightarrow a}} f(x) = f(a)\) holds true. This becomes particularly useful in calculus as it allows for a smooth evaluation of limits at points within the domain of the function.

An everyday example would be driving a car at a constant speed; as you predict where you will be after a certain time, there should be no surprises, much like how continuous functions behave.

Limits are an essential part of understanding the behavior of functions, especially as they approach a certain value. In the realm of calculus, several limit properties allow mathematicians to efficiently compute or manipulate limits without tediously evaluating the function at every nearby point.

Here are some key limit properties that are particularly useful:

• The limit of a constant is just the constant itself.
• The limit of a sum is the sum of the limits (if they exist).
• For polynomial functions, like cubic functions, the limit can be found by simply substituting the point into the function (thanks to their continuity).
• Limits can be distributed over multiplication and division, given certain conditions.

With these properties, tackling limits becomes a systematic task rather than a daunting challenge, allowing students to break down complex limit problems into manageable pieces.

Cubic functions have the form \(f(x) = ax^{3} + bx^{2} + cx + d\), where 'a', 'b', 'c', and 'd' are constants and \(a eq 0\). As the name suggests, the highest power in a cubic function is 3, which gives the function its characteristic 'S' shaped curve when graphed.

One of the key characteristics of cubic functions is their continuity; they are smooth and unbroken over the entire range of real numbers. This means, as shown in the exercise, that limits involving cubic functions are straightforward to compute. Another interesting fact about cubic functions is that they can have zero, one, two, or three real roots, depending on their coefficients.

Cubic functions also tend to display interesting behavior such as inflection points, which are points where the curvature of the graph changes direction. Understanding the nature of cubic functions enhances one's abilities to analyze them both algebraically and graphically.