How Do Limits Behave for Piecewise Functions at Specific Points?

Thread starter nycmathguy
Start date Jun 17, 2021

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SUMMARY

This discussion focuses on the behavior of limits for the piecewise function defined as f(x) = 1 if x is an integer and f(x) = 0 if x is not an integer. The limits investigated include lim f(x) as x approaches 2, 1/2, 3, and 0. The conclusions drawn are that lim f(x) as x approaches 2 is 1, lim f(x) as x approaches 1/2 is 0, lim f(x) as x approaches 3 is 1, and lim f(x) as x approaches 0 is 0. The distinction between the function value at a point and the limit as x approaches that point is emphasized throughout the discussion.

PREREQUISITES

Understanding of piecewise functions
Knowledge of limits in calculus
Familiarity with integer and non-integer values
Ability to differentiate between function values and limit values

NEXT STEPS

Study the properties of piecewise functions in depth
Learn about continuity and discontinuity in functions
Explore limit definitions and their applications in calculus
Investigate the behavior of limits at points of discontinuity

USEFUL FOR

Students studying calculus, particularly those focusing on limits and piecewise functions, as well as educators looking to clarify these concepts for their students.

Jun 19, 2021

nycmathguy

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.