Continuous Extension to a Point (Factoring Question)

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Homework Help Overview

The discussion revolves around defining a function f(6) to ensure continuity at s=6, specifically focusing on the need to factor expressions to resolve a zero-denominator issue when calculating limits.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the necessity of factoring s^2-36 from the numerator to evaluate the limit as s approaches 6. There is an exploration of the correct factors to use, with one participant noting the presence of a linear factor instead.

Discussion Status

Some participants have provided guidance on factoring, with one confirming the successful identification of the correct factor (s-6) to cancel. There is an acknowledgment of initial confusion regarding the factors involved.

Contextual Notes

Participants are navigating the constraints of the problem, particularly the requirement to maintain continuity at a specific point and the implications of zero-denominator scenarios in limit calculations.

BraedenP
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Homework Statement



Define f(6) in a way that extends
gif.latex?f(s)=\frac{s^3-216}{s^2-36}.gif
to be continuous at s=6.

Homework Equations



None. Only limits are required.

The Attempt at a Solution



In order to figure out which point needs to be added to the function, I have to find the limit of this function as s->6. This will result in a zero-denominator, however, so I need to factor [PLAIN]http://latex.codecogs.com/gif.latex?s^2-36 out of the numerator, so that I can cancel them and use the resulting polynomial to calculate the limit.

Basically, I just need help factoring [PLAIN]http://latex.codecogs.com/gif.latex?s^2-36 out of [URL]http://latex.codecogs.com/gif.latex?s^3-216;[/URL] I can solve the rest of the question easily.

Help would be appreciated.

Thanks,
Braeden
 
Last edited by a moderator:
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BraedenP said:

Homework Statement



Define f(6) in a way that extends
gif.latex?f(s)=\frac{s^3-216}{s^2-36}.gif
to be continuous at s=6.

Homework Equations



None. Only limits are required.

The Attempt at a Solution



In order to figure out which point needs to be added to the function, I have to find the limit of this function as s->6. This will result in a zero-denominator, however, so I need to factor [PLAIN]http://latex.codecogs.com/gif.latex?s^2-36 out of the numerator, so that I can cancel them and use the resulting polynomial to calculate the limit.

Basically, I just need help factoring [PLAIN]http://latex.codecogs.com/gif.latex?s^2-36 out of [URL]http://latex.codecogs.com/gif.latex?s^3-216;[/URL] I can solve the rest of the question easily.
The numerator doesn't have a factor of s2 - 36. There is a linear factor that can be pulled out, however.
 
Last edited by a moderator:
Mark44 said:
The numerator doesn't have a factor of s2 - 36. There is a linear factor that can be pulled out, however.

Yeah, I figured it out, thanks. I'm not quite sure what I was trying to do. I ended up factoring (x-6) out of the numerator and denominator and cancelling those.
 
Hopefully, you factored out s - 6 rather than x - 6.
 
Mark44 said:
Hopefully, you factored out s - 6 rather than x - 6.

Oh yeah.. sorry. I factored (s-6)
 

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