Use the graph of f(x) to investigate the limit

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Homework Help Overview

The discussion revolves around investigating the limit of a piecewise-defined function f(x) as x approaches a specific value c, particularly focusing on the behavior of the function from both the left-hand side and right-hand side of c.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants examine the limits from both sides of c, noting discrepancies between left-hand and right-hand limits. There is a focus on the implications of piecewise definitions, particularly regarding the inclusion of equal signs.

Discussion Status

The discussion includes various interpretations of the piecewise function's definitions and their impact on the limit. Some participants affirm the correctness of the limit analysis, while others raise questions about the definitions used. There is no explicit consensus, but guidance has been offered regarding the correctness of the one-sided limits and the conclusion about the two-sided limit.

Contextual Notes

Participants are navigating the nuances of piecewise functions and their definitions, with specific attention to how these definitions affect limit calculations. The original poster's conclusions about the limits are reiterated, but the discussion remains open to further exploration of similar problems.

nycmathguy
Homework Statement
Use the graph of f(x) to investigate the limit.
Relevant Equations
Piecewise-defined Function
Use the graph to investigate the limit of f(x) as x tends to c at the number c.

See attachments.

Based on the graph of f(x), here is what I did:

lim (2x + 1) as x tends to 0 from the left is 1.

lim (2x) as x tends to 0 from the right is 0.

LHL does not equal RHL.

I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
 

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nycmathguy said:
Homework Statement:: Use the graph of f(x) to investigate the limit.
Relevant Equations:: Piecewise-defined Function

I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
Um, one of your piecewise continuous definitions includes and "=" sign...
 
nycmathguy said:
I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
Looks good to me.
 
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berkeman said:
Um, one of your piecewise continuous definitions includes and "=" sign...
Meaning?
 
Mark44 said:
Looks good to me.

That's what I thought. I may post one more like this involving a piecewise function in terms of 3 pieces. Let me think about it.
 
berkeman said:
Um, one of your piecewise continuous definitions includes and "=" sign...
nycmathguy said:
Meaning?
Doesn't matter whether one or both or neither of the function definitions has an = in it. The one-sided limits you found are correct, and your conclusion that the two-sided limit doesn't exist is also correct.

Well done.
 
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Mark44 said:
Doesn't matter whether one or both or neither of the function definitions has an = in it. The one-sided limits you found are correct, and your conclusion that the two-sided limit doesn't exist is also correct.

Well done.

Beautiful. I may post one more but this time, the piecewise function will be three pieces. Stay tune.
 

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