 Nickolas Taylor

2022-07-09

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

I understand I have to rewrite the fraction somehow for the denominator not to equal 0, but I don't know where to start. Brendan Bush

Expert

Using the Euclidean division of ${x}^{3}+4x-5$ by $x-1$ we get
$f\left(x\right)=\frac{{x}^{3}+4x-5}{{x}^{2}-1}=\frac{\left(x-1\right)\left({x}^{2}+x+5\right)}{\left(x-1\right)\left(x+1\right)}$ Cooper Doyle

Expert

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
$f\left(x\right)=ax+b+\frac{cx+d}{{x}^{2}-1}$
for some constants $a,b,c,d,$ where the linear numerator $cx+d$ 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
${x}^{3}+4x-5=\left(ax+b\right)\left({x}^{2}-1\right)+cx+d.$
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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