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

Nickolas Taylor

Nickolas Taylor

Answered

2022-07-09

Rewrite fraction to calculate limit
I 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.

Answer & Explanation

Brendan Bush

Brendan Bush

Expert

2022-07-10Added 14 answers

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

Cooper Doyle

Expert

2022-07-11Added 2 answers

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've found these, the rest should fall right out of your usual limit manipulations.

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