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Understanding the concept of limits is crucial when exploring the continuity and behavior of functions at certain points. A limit essentially answers the question: as we approach a certain value of x, what value does the function approach? Let's consider what happens as we approach the number 0 from both the left (\(x \to 0^-\)) and the right (\(x \to 0^+\)) for our function.

When we compute the limit from the left side, we use the expression that defines the function for values less than or equal to zero; in the provided exercise, this is the linear function \(f(x)=x+2\). Conversely, to find the limit from the right side, we use the definition of the function for values greater than zero; for our function, this is the linear expression \(f(x)=2-3x\). If these two limits are equal, as they are in our example where both are 2, it suggests that the function approaches the same value from both sides at \(x=0\). This is a core indicator that our function is continuous at that particular point.

Our focus function is an example of a piecewise function, which is constructed from multiple segments, each defined over a certain interval. The function changes its formula depending on the value of x. In this case, the function \(f(x)\) is defined as \(x+2\) when \(x \leq 0\) and as \(2-3x\) when \(x>0\).

One key aspect of understanding piecewise functions is examining how these 'pieces' interact at their boundaries. In order to be continuous, a piecewise function must not have any breaks or jumps at these boundary points. In our example, the boundary point at \(x=0\) is smooth because the value of both 'pieces' is 2; this seamless transition ensures continuity. To enhance learning, visual aids such as graphing the function can greatly assist in grasping the behavior at the boundary points, where students can visually verify the continuous nature of the function at \(x=0\).

When examining differentiability, we delve into the concept of derivatives—the rate at which a function is changing at any point. If a function is differentiable at a point, it means it has a well-defined tangent with a certain slope at that point. For our function \(f(x)\), we differentiate both pieces where \(f'(x)\) represents the derivative of \(f\).

The derivative \(f'(x)\) within the interval \(x \leq 0\) is a constant 1, indicating a steady, unchanging rate. However, the derivative for \(x > 0\) is -3, implying a different rate of change. At the boundary between these intervals, \(x=0\), if the rates do not match, we encounter what's called a 'corner' or 'cusp' in the graph of the function, indicating a point of non-differentiability. This is precisely what happens in our exercise, where the left and right derivatives do not match, resulting in a function that is continuous but not differentiable at \(x=0\). Using illustrative graphs to show the different slopes on each side can be especially helpful for students to visualize why there's no single tangent line at \(x=0\).