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Analyzing Function Behavior: Limits, Continuity, and Differentiability

Explore the characteristics of a given function to determine points where the limit does not exist, where the limit from the left fails, and where the function is neither continuous nor differentiable. Additionally, we will evaluate the derivative f’(x) for specific intervals and values, including f’(x) for x < 1 and for -1 < x < 1. Understanding these aspects is crucial for analyzing the behavior of functions in calculus.

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Analyzing Function Behavior: Limits, Continuity, and Differentiability

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  1. At how many points for the given function does the limit not exist?

  2. At how many points for the given function does the limit from the left not exist?

  3. At how many points is the given function not continuous ?

  4. At how many points is the given function not differentiable ?

  5. For the given function, what is f’(x) for x < 1? • 1 • x • 2x • 0 • x+1

  6. For the given function, what is f’(x) for -1< x < 1? • 1 • x • 2x • 0 • x+1

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