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Limits help us understand the behavior of functions as they approach a certain point. Imagine driving towards a stop sign. The limit describes what happens to your speed as you get closer to the stop sign.
In mathematical terms, the limit of a function \(f(x)\) as \(x\) approaches some value \(c\) is written as \(\lim_{{x \to c}} f(x)\). This tells us the value that \(f(x)\) gets closer to as \(x\) gets closer to \(c\).
There are two types of limits—left-hand limit and right-hand limit. When \(x\) approaches \(c\) from the left (or smaller values), it's called the left-hand limit, written as \(\lim_{{x \to c^-}} f(x)\). When \(x\) approaches \(c\) from the right (or larger values), it's called the right-hand limit, written as \(\lim_{{x \to c^+}} f(x)\).

Piecewise functions are those defined by different expressions in different intervals of their domain. Think of a road that changes speed limits in different sections. Likewise, a piecewise function uses different equations in different parts of its domain.
For example, the piecewise function given in the exercise is defined as follows:

• For \(x < 3\): \(f(x) = \frac{1}{3}x + 4\)
• For \(x \geq 3\): \(f(x) = 2x - 1\)

Each part of the function applies to a different region of the \(x\) values. This makes it tricky to analyze at the points where the definition changes, like at \(x = 3\) in this exercise.

A function is continuous if you can draw its graph without lifting your pencil. More formally, a function \(f(x)\) is continuous at a point \(x = c\) if the following three conditions are satisfied:

• The limit of \(f(x)\) as \(x\) approaches \(c\) from the left exists and is equal to \(L\): \(\lim_{{x \to c^-}} f(x) = L\)
• The limit of \(f(x)\) as \(x\) approaches \(c\) from the right exists and is equal to \(L\): \(\lim_{{x \to c^+}} f(x) = L\)
• The value of the function at \(x = c\) exists and is equal to \(L\): \(f(c) = L\)

In our exercise, to prove that the function is continuous at \(x = 3\), we checked these three things. Both the left-hand and right-hand limits as \(x\) approached 3 were 5, and the value of the function at \(x = 3\) was also 5. Therefore, the function is continuous at \(x = 3\).

Evaluating limits involves calculating the value that \(f(x)\) approaches as \(x\) gets closer to a certain point. For piecewise functions, it's important to check the limit from both sides of the point.
In the exercise:

• We first calculated the left-hand limit as \(x\) approached 3 using the expression \(\frac{1}{3}x + 4\). We found \(\lim_{{x \to 3^-}} f(x) = 5\).
• Next, we calculated the right-hand limit from the expression \(2x - 1\), giving \(\lim_{{x \to 3^+}} f(x) = 5\).
• Then, we checked the value of \(f(x)\) at \(x = 3\), which was also 5.

Since the left-hand limit, right-hand limit, and \(f(3)\) are all equal, the limit exists and the function is continuous at \(x = 3\). Evaluating limits in steps helps ensure all conditions for continuity are met.