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The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular point. When we talk about \(\lim_{x \to c} f(x)\), we are examining what value the function f(x) gets close to as x approaches the number c from either side. A key point here is that the function doesn't necessarily have to reach or even include c to have a limit at that point.

In the problem, we calculated \(\lim_{x \to 1} f(x)\). The function \( f(x) \) is piecewise, meaning it has distinct rules for different intervals of x. We found that as x approaches 1 from the left or right, the function values approach 2. However, the limit is not defined because the piecewise rule at \(x = 1\) gives a value of 1 instead.

For functions to have limits at a certain point, both left and right limits must exist and equal each other. Only then do we say the function has a definite limit at that point. If they differ, or if the function is not continuous at that point, like in this case at x = 1, we say the limit does not exist.

Graph sketching is a critical skill in understanding how functions behave visually, allowing us to glean insights about limits and continuity. Seeing a graph helps you to apply concepts from calculus, like limits, more effectively. With piecewise functions, narratives are even clearer as each piece of the function is plotted according to its rule across its interval.

For our problem, \(f(x) = \begin{cases} x^2 + 1 & \text{if } x < 1 \ 1 & \text{if } x = 1 \ x + 1 & \text{if } x > 1 \end{cases}\), graphing involves plotting:

• \(y = x^2 + 1\) as a parabola opening upwards for \(x < 1\).
• A single dot at (1,1) for the case where x equals 1.
• The line \(y = x + 1\) for \(x > 1\), initiating just after x = 1.

Graph sketching reveals the general slope of the curve, the points of discontinuity, and the behavior near specific x-values, all essential for understanding limits.

Continuity is a key concept in calculus and refers to a function being unbroken or smooth at a particular point or over an interval. A function is continuous at a point \(c\) if:

• The function value \(f(c)\) is defined.
• The limit \(\lim_{x \to c} f(x)\) exists.
• The limit \(\lim_{x \to c} f(x)\) equals \(f(c)\).

If any one of these conditions is not met, the function is not continuous at that point. In our exercise, the function is defined to be 1 at \(x = 1\), yet the left and right limits approach 2, not 1. Therefore, the function is discontinuous at that point.

Understanding continuity through a graph or through limits allows us to better predict and understand the behavior of functions in important mathematical and real-world contexts. Graph discontinuities often manifest clearly as gaps, jumps, or holes, marking where the function fails to be continuous.