Rewrite fraction to calculate limitI am practising finding limits. However, I can't seem to figure out this one. f ( x ) = x 3 + 4 x − 5 x 2 − 1 as x goes to 1 I understand I have to rewrite the fraction somehow for the denominator not to equal 0, but I don't know where to start.

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Rewrite fraction to calculate limitI am practising finding limits. However, I can&#039;t seem to figure out this one.  f  (  x  )  =                    x        3            +      4      x      −      5                      x        2            −      1             as           x         goes to           1      I understand I have to rewrite the fraction somehow for the denominator not to equal 0, but I don&#039;t know where to start.

Brendan Bush · Accepted Answer

Using the Euclidean division of        x    3    +  4  x  −  5 by   x  −  1 we get  f  (  x  )  =                    x        3            +      4      x      −      5                      x        2            −      1        =            (      x      −      1      )      (              x        2            +      x      +      5      )              (      x      −      1      )      (      x      +      1      )

Cooper Doyle · Accepted Answer

One idea is to use polynomial long division.The idea is to note that you have a cubic divided by a quadratic, so the degree of the numerator is greater by 1. Consequently, we can conclude that  f  (  x  )  =  a  x  +  b  +            c      x      +      d                      x        2            −      1      for some constants   a  ,  b  ,  c  ,  d  ,  where the linear numerator   c  x  +  d is to allow for the fact that there may be a remainder term, which is necessarily of lower degree than the denominator.Multiplying both sides of this equation by       x    2    −  1 -which is non-0 for x sufficiently close (but not equal) to 1--we obtain      x    3    +  4  x  −  5  =  (  a  x  +  b  )  (      x    2    −  1  )  +  c  x  +  d  .Expand the product on the right-hand side to give yourself a system of equations. Solve for   a  ,  b  ,  c  ,  d  . Once you&#039;ve found these, the rest should fall right out of your usual limit manipulations.

Rewrite fraction to calculate limit I am practising finding limits. However, I can't seem to figure

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