I'm learning about holes in rational functions in precalculus, and I'm confused as to why they exist, having some knowledge of limits. Say I have a function R ( x ) = ( x − 5 ) ( x + 2 ) ( x + 4 ) ( x + 2 ) . If I were to find where the hole lies for this function, I would plug −2 into x − 5 x + 4 which would give me a hole at ( − 2 , − 7 2 ). However, that turns out to be the limit of R ( x ) (using L'Hopital's Rule): lim x → − 2 ( x − 5 ) ( x + 2 ) ( x + 4 ) ( x + 2 ) = lim x → − 2 x 2 − 3 x − 10 x 2 + 6 x + 8 = lim x → − 2 2 x − 3 2 x + 6 = − 7 2 So why is there a hole in the graph if the limit exists at ( − 2 , − 7 2 )? Am I misunderstanding the definition of a limit? Thanks.

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I&#039;m learning about holes in rational functions in precalculus, and I&#039;m confused as to why they exist, having some knowledge of limits. Say I have a function   R  (  x  )  =            (      x      −      5      )      (      x      +      2      )              (      x      +      4      )      (      x      +      2      )      . If I were to find where the hole lies for this function, I would plug −2 into             x      −      5              x      +      4       which would give me a hole at   (  −  2  ,  −      7    2    ). However, that turns out to be the limit of   R  (  x  ) (using L&#039;Hopital&#039;s Rule):      lim          x      →      −      2                  (      x      −      5      )      (      x      +      2      )              (      x      +      4      )      (      x      +      2      )        =      lim          x      →      −      2                          x        2            −      3      x      −      10                      x        2            +      6      x      +      8        =      lim          x      →      −      2                  2      x      −      3              2      x      +      6        =  −      7    2  So why is there a hole in the graph if the limit exists at   (  −  2  ,  −      7    2    )? Am I misunderstanding the definition of a limit? Thanks.

talhekh · Accepted Answer

The domain of definition of the function   R is the set of all reals   x such that   (  x  +  4  )  (  x  +  2  ) is not equal to 0, to avoid division by 0 when evaluating. This is the definition or a convention if you prefer. (One could do things differently, too, but this is what was agreed upon.)It is true that the limit you mention exists. Therefore you could define a new function   Q  (  x  ) that agrees with   R  (  x  ) everywhere where   R is defined and in addition   Q  (  −  2  )  =  −  7      /    2, and this function would be a continuous function.Thus this type of &quot;hole&quot; can be plugged, and one thus calls it a removable &quot;hole&quot; (or a removable singularity). By contrast, the &quot;hole&quot; at −4 cannot be plugged in this way.

I'm learning about holes in rational functions in precalculus, and I'm confused as to why they exist

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