How Do Limits Behave for Piecewise Functions at Specific Points?

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The discussion focuses on understanding limits of a piecewise function at specific points, particularly at x=2 and x=1/2. Participants clarify that the function f(x) equals 1 for integers and 0 for non-integers, leading to the conclusion that the limit as x approaches 2 is 1, while the limit as x approaches 1/2 is 0. Further exploration of limits at x=3 and x=0 reinforces that limits depend on the behavior of the function as it approaches a point, rather than the function's value at that point. The conversation emphasizes the distinction between limit values and function values, especially for discontinuous functions. Understanding these concepts is crucial for tackling future calculus problems involving limits.
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Mark44 said:
So make sure you understand the difference between, say, ##f(c)## and ##\lim_{x \to c} f(x)##. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.
Will do.
 
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