Finding removable and jump discontinuities

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Homework Help Overview

The discussion revolves around identifying removable and jump discontinuities in a piecewise function defined differently for values less than, equal to, and greater than 2. The function involves parameters that influence its continuity at the point x = 2.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the conditions for removable discontinuities, specifically focusing on the limits from both sides of x = 2 in relation to the parameter b. There is uncertainty about the nature of the limits and whether they approach infinity.

Discussion Status

The conversation is ongoing, with participants attempting to clarify the limits involved and how they relate to the parameter b. Some guidance has been offered regarding the need to equate the limits for removable discontinuities, but there is still confusion about the limits themselves.

Contextual Notes

There is a lack of consensus on the behavior of the limits as x approaches 2, with some participants questioning the values and others attempting to clarify the definitions of removable and jump discontinuities.

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Homework Statement



f(x) = -x + b, if x < 2
= 5, if x = 2
= -20/(x-b) + 1, if x > 2

For what value(s) of b does f have a removable discontinuity at 2?
For what value(s) of b does f have a (finite) jump discontinuity at 2? Write your answer in interval notation.

The Attempt at a Solution



I'm completely stumped on the removable discontinuity, because I thought you had to be able to cancel out the bottom?

And I'm not sure how to find a jump discontinuity.
 
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A removable discontinuity is a value of b where the limit as x->2 from below and the limit as x->2 from above are equal. Your first job is to find those two limits in terms of b. Then equate them. Can you do that?
 
Are the limits not -inf and inf?
 
DanielJackins said:
Are the limits not -inf and inf?

No. The limits as x->2! The limit from above is (x>2) is -20/(2-b)+1, isn't it? What's the limit from below?
 

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