Prove that the limit x tends to 1 (2x^4-6x^3+x^2+3)÷(x-1)=-8

First note that if , you have



Now, you're looking for such that for any , you have



Note that you can divide through the left side of the inequality by once more:



So it follows that you need to find an appropriate that will guarantee



For the moment, let's fix . Then by this assumption, we have



From this we get







where the upper bound is what we care about. With this assumption, we then get that



which suggests that can be taken to be either the smaller of 1 or , or , to guarantee that the function gets arbitrarily close to -8.